System of polynomial equations

In ring theory and Frobenius algebra extensions, fields of mathematics, there is a notion of depth two subring or depth of a Frobenius extension. The notion of depth two is important in a certain noncommutative Galois theory, which generates Hopf algebroids in place of the more classical Galois groups, whereas the notion of depth greater than two measures the defect, or distance, from being depth two in a tower of iterated endomorphism rings above the subring. A more recent definition of depth of any unital subring in any associative ring is proposed (see below) in a paper studying the depth of a subgroup of a finite group as group algebras over a commutative ring.

Definition and first examples

A unital subring $B\subseteq A$ has (or is) right depth two if there is a split epimorphism of natural A-B-bimodules from $A^n\to A{\otimes }_{B}A$ for some positive integer n; by switching to natural B-A-bimodules, there is a corresponding definition of left depth two. Here we use the usual notation $A^n=A\times \dots \times A$ (n times) as well as the common notion, p is a split epimorphism if there is a homomorphism q in the reverse direction such that pq = identity on the image of p. (Sometimes the subring B in A is referred to as the ring extension A over B; the theory works as well for a ring homomorphism B into A, which induces right and left B-modules structures on A.) Equivalently, the condition for left or right depth two may be given in terms of a split monomorphism of bimodules where the domains and codomains above are reversed.

For example, let A be the group algebra of a finite group G (over any commutative base ring k; see the articles on group theory and group ring for the elementary definitions). Let B be the group (sub)algebra of a normal subgroup H of index n in G with coset representatives $g_1,\cdots ,g_n$. Define a split A-B epimorphism p:$A^n\to A{\otimes }_{B}A$ by $p((a_1,\cdots ,a_n))=\sum _{i=1}^n{a}_i{g}_i^{-1}{\otimes }_B{g}_i$. It is split by the mapping $q:A{\otimes }_{B}A\to A^n$ defined by $q(a{\otimes }_B{a}^{\prime })=(a{\gamma }_1(a^{\prime }),\cdots ,a{\gamma }_n(a^{\prime }))$ where ${\gamma }_i(g)={\delta }_{ij}g$ for g in the coset $g_{j}H$ (and extended linearly to a mapping A into B, a B-B-module homomorphism since H is normal in G): the splitting condition pq = the identity on $A{\otimes }_{B}A$ is satisfied. Thus B is right depth two in A.

As another example (perhaps more elementary than the first; see ring theory or module theory for some of the elementary notions), let A be an algebra over a commutative ring B, where B is taken to be in the center of A. Assume A is a finite projective B-module, so there are B-linear mapping $f_i:A\to B$ and elements $x_i\in A$ (i = 1,...,n) called a projective base for the B-module A if it satisfies $\sum _{i=1}^n{x}_i{f}_i(a)=a$ for all a in A. It follows that B is left depth two in A by defining $p(a_1,\cdots ,a_n)=\sum _{i=1}^n{x}_i{\otimes }_B{a}_i$ with splitting map $q(a{\otimes }_B{a}^{\prime })=(f_1(a)a^{\prime },\cdots ,f_n(a)a^{\prime })$ as the reader may verify. A similar argument naturally shows that B is right depth two in A.

Depth in relation to Hopf algebras

For a Frobenius algebra extension A | B (such as A and B group algebras of a subgroup pair of finite index) the two one-sided conditions of depth two are equivalent, and a notion of depth n > 2 makes sense via the right endomorphism ring extension iterated to generate a tower of rings (a technical procedure beyond the scope of this survey, although the first step, the endomorphism ring theorem, is described in the section on Frobenius extension under Frobenius algebra). For example, if B is a Hopf subalgebra of a finite dimensional Hopf algebra, then B has depth two in A if and only if B is normal in A (i.e. invariant under the left and right adjoint actions of A). Since a group algebra is a Hopf algebra, the first example above illustrates the back implication of the theorem. Other examples come from the fact that finite Hopf-Galois extensions are depth two in a strong sense (the split epimorphism in the definition may be replaced by a bimodule isomorphism).

Let R be a Hopf subalgebra of a finite-dimensional Hopf algebra H. Let R° denote the maximal ideal of elements of R having counit value 0. Then R°H is a right ideal and coideal in H, and the quotient module V = H/R°H is a right H-module coalgebra. For example, if H is a group algebra, then R is a subgroup algebra of H, and one shows as an exercise that V is isomorphic to the space of right cosets. The 2013 paper referenced below proves that the depth of R in H is determined to the nearest even value by the depth of V as an R-module (by restriction). The depth of V as an R-module is defined in that paper to be the least positive integer n such that V⊗⋅⋅⋅⊗V (n times V, tensor product of R-modules, diagonal action) has the same constituent indecomposable modules as V ⊗⋅⋅⋅⊗V (n+1 times V) (not counting multiplicities). As a consequence, the depth of R in H is finite if and only if its "generalized quotient module" V represents an algebraic element in the representation ring (or Green ring) of R. This is the case for example if V is a projective module, or if V is a permutation module over a group algebra R (i.e., V has a basis that is a G-set).

Depth in relation to finite dimensional semisimple algebras and subgroups of finite groups

If M is the inclusion matrix (or incidence matrix of the Bratteli diagram) of finite dimensional semisimple (complex) algebras B and A, the depth two condition on the subalgebra B in A is given by an inequality $M{M}^{t}M\le nM$ for some positive integer n (and each corresponding entry). Denoting the left-hand side of this inequality by the power $M^3$ and similarly for all powers of the inclusion matrix M, the condition of being depth $m\ge 1$ on the subalgebra pair of semisimple algebras is: $M^{m+1}\le n{M}^{m-1}$. (Notice that if M satisfies the depth m condition, then it satisfies the depth m+1 condition.) For example, a depth one subgroup H of a finite group G, viewed as group algebras CH in CG over the complex numbers C, satisfies the condition on the centralizer $G=H{C}_G(X)$ for each cyclic subgroup X in H (whence normal) ; e.g. H a subgroup in the center of G, or G = H x K. As another example, consider the group algebras $B=C{S}_2$ and $A=C{S}_3$, the order 2 and order 6 permutation groups on three letter a,b,c where the subgroup fixes c. The inclusion matrix may be computed in at least three ways via idempotents, via character tables or via Littlewood-Richardson coefficients and combinatorics of skew tableaux to be (up to permutation) the 2 by 3 matrix with top row 1,1,0 and bottow row 0,1,1, which has depth three after applying the definition.

In a 2011 article in the Journal of Algebra by R. Boltje, S. Danz and B. Kuelshammer, they provide a simplified and extended definition of the depth of any unital subring B of associative ring A to be 2n+1 if $A{\otimes }_B\cdots {\otimes }_{B}A$ (n+1 times A) is isomorphic to a direct summand in $\underset{i=1}{\overset{m}{\oplus }}A{\otimes }_B\cdots {\otimes }_{B}A$ (n times A) as B-B-bimodules for some positive integer m; similarly, B has depth 2n in A if the same condition is satisfied more strongly as A-B-bimodules (or equivalently for free Frobenius extensions, as B-A-bimodules). (This definition is equivalent to an earlier notion of depth in case A is a Frobenius algebra extension of B with surjective Frobenius homomorphism, for example A and B are complex semisimple algebras.) Again notice that a subring having depth m implies that it has depth m+1, so they let $d(B,A)$ denote the minimal depth. They then apply this to the group algebras of G and H over any commutative ring R. They define a combinatorial depth $d_c(H,G)$ of a subgroup H of a finite group G mimicking the definition of depth of a subring but using G-sets and G-set homomorphisms instead of modules and module homomorphisms. They characterize combinatorial depth n as a condition on the number of conjugates of H intersecting in G thereby showing that combinatorial depth is finite. The depth d(RH,RG)) is shown to be bounded by $d_c(H,G)$ which in turn is bounded by twice the index of the normalizer of H in G.

Galois theory for depth two extensions

Main classes of examples of depth two extensions are Galois extensions of algebras being acted upon by groups, Hopf algebras, weak Hopf algebras or Hopf algebroids (types of quantum algebras defined in the article on Hopf algebra); for example, suppose a finite group G acts by automorphisms on an algebra A, then A is a depth two extension of its subalgebra B of invariants if the action is G-Galois, explained in detail in the article on Frobenius algebra extension.

Conversely, any depth two extension A | B has a Galois theory based on the natural action of $\text{End}{}_B{A}_B$ on A: denoting this endomorphism ring by S, one shows S is a left bialgebroid over the centralizer R (those a in A commuting with all b in B) with a Galois theory similar to that of Hopf-Galois theory. There is a right bialgebroid structure on the B-centralized elements T in $A{\otimes }_{B}A$ dual over R to S; certain endomorphism rings decompose as smash product, such as $\text{End}{A}_B\cong A{\otimes }_{R}S$, i.e. isomorphic as rings to the smash product of the bialgebroid S (or its dual) with the ring A it acts on. Something similar is true for T and $\text{End}A{\otimes }_B{A}_A$ (often called a theory of duality, which began in operator algebras in the 1970s). If A | B is in addition to being depth two a Frobenius algebra extension, the right and left endomorphism rings are anti-isomorphic, which restricts to an antipode on the bialgebroid $\text{End}{}_B{A}_B$ satisfying axioms of a Hopf algebroid (see the article on Hopf algebra under related concepts). There is the following relation with relative homological algebra: the relative Hochschild complex of A over B with coefficients in A, and cup product, is isomorphic as differential graded algebras to the Amitsur complex of the R-coring S (with group-like element the identity on A; see Brzezinski-Wisbauer for the definition of the Amitsur cochain complex with product).

The Galois theory of a depth two extension is not irrelevant to a depth n > 2 Frobenius extension since such a depth n extension embeds in a depth two extension in a tower of iterated endomorphism rings. For example, given a depth three Frobenius extension of ring A over subring B, one can show that the left multiplication monomorphism $\lambda :B\to \text{End}{A}_B,\lambda (b)(a)=ba$ has depth two.

References

• Tomasz Brzezinski; Robert Wisbauer, Corings and Comodules. London Math. Soc. Lect. Note Ser., 309. Cambridge University Press, 2003. ISBN 0-521-53931-5

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  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010