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In [[mathematics]], especially in the field of [[group theory]], a '''divisible group''' is an [[abelian group]] in which every element can, in some sense, be divided by positive integers, or more accurately, every element is an ''n''th multiple for each positive integer ''n''. Divisible groups are important in understanding the structure of abelian groups, especially because they are the [[injective module|injective]] abelian groups. | |||
==Definition== | |||
An abelian group ''G'' is '''divisible''' if, for every positive integer ''n'' and every ''g'' in ''G'', there exists ''y'' in ''G'' such that ''ny'' = ''g''.<ref>Griffith, p.6</ref> An equivalent condition is: for any positive integer ''n'', ''nG'' = ''G'', since the existence of ''y'' for every ''n'' and ''g'' implies that ''nG'' ⊇ ''G'', and in the other direction ''nG'' ⊆ ''G'' is true for every group. A third equivalent condition is that an abelian group ''G'' is divisible if and only if ''G'' is an [[injective object]] in the [[category of abelian groups]]; for this reason, a divisible group is sometimes called an '''injective group'''. | |||
An abelian group is ''p''-'''divisible''' for a [[prime number|prime]] ''p'' if for every positive integer ''n'' and every ''g'' in ''G'', there exists ''y'' in ''G'' such that ''p''<sup>''n''</sup>''y'' = ''g''. Equivalently, an abelian group is ''p''-divisible if and only if ''pG'' = ''G''. | |||
==Examples== | |||
* The [[rational number]]s <math>\mathbb Q</math> form a divisible group under addition. | |||
* More generally, the underlying additive group of any [[vector space]] over <math>\mathbb Q</math> is divisible. | |||
* Every [[quotient group|quotient]] of a divisible group is divisible. Thus, <math>\mathbb Q/\mathbb Z</math> is divisible. | |||
* The ''p''-[[primary component]] <math>\mathbb Z[1/p]/\mathbb Z</math> of <math>\mathbb Q/ \mathbb Z</math>, which is isomorphic to the ''p''-[[quasicyclic group]] <math>\mathbb Z[p^\infty]</math> is divisible. | |||
* The multiplicative group of the [[complex number]]s <math>\mathbb C^*</math> is divisible. | |||
* Every [[existentially closed]] group (in the [[model theory|model theoretic]] sense) is divisible. | |||
* The space of [[Orientation (vector space)|orientation]]-preserving [[isometries]] of <math>\mathbb R^2</math> is divisible. This is because each such isometry is either a translation or a rotation about a point, and in either case the ability to "divide by n" is plainly present. This is the simplest example of a non-[[Abelian group|Abelian]] divisible group. | |||
==Properties== | |||
* If a divisible group is a subgroup of an abelian group then it is a [[direct summand]].<ref>Hall, p.197</ref> | |||
* Every abelian group can be embedded in a divisible group.<ref>Griffith, p.17</ref> | |||
* Non-trivial divisible groups are not [[finitely generated abelian group|finitely generated]]. | |||
* Further, every abelian group can be embedded in a divisible group as an [[essential subgroup]] in a unique way.<ref>Griffith, p.19</ref> | |||
* An abelian group is divisible if and only if it is ''p''-divisible for every prime ''p''. | |||
* Let <math>A</math> be a ring. If <math>T</math> is a divisible group, then <math>\mathrm{Hom}_\mathbf{Z} (A,T)</math> is injective in the category of <math>A</math>-modules.<ref>Lang, p. 106</ref> | |||
==Structure theorem of divisible groups== | |||
Let ''G'' be a divisible group. Then the [[torsion subgroup]] Tor(''G'') of ''G'' is divisible. Since a divisible group is an [[injective module]], Tor(''G'') is a [[direct summand]] of ''G''. So | |||
:<math>G = \mathrm{Tor}(G) \oplus G/\mathrm{Tor}(G).</math> | |||
As a quotient of a divisible group, ''G''/Tor(''G'') is divisible. Moreover, it is [[torsion (algebra)|torsion-free]]. Thus, it is a vector space over '''Q''' and so there exists a set ''I'' such that | |||
:<math>G/\mathrm{Tor}(G) = \oplus_{i \in I} \mathbb Q = \mathbb Q^{(I)}.</math> | |||
The structure of the torsion subgroup is harder to determine, but one can show{{Citation needed|date=October 2011}} that for all [[prime number]]s ''p'' there exists <math>I_p</math> such that | |||
:<math>(\mathrm{Tor}(G))_p = \oplus_{i \in I_p} \mathbb Z[p^\infty] = \mathbb Z[p^\infty]^{(I_p)},</math> | |||
where <math>(\mathrm{Tor}(G))_p</math> is the ''p''-primary component of Tor(''G''). | |||
Thus, if '''P''' is the set of prime numbers, | |||
:<math>G = (\oplus_{p \in \mathbf P} \mathbb Z[p^\infty]^{(I_p)}) \oplus \mathbb Q^{(I)}.</math> | |||
==Injective envelope== | |||
{{main|Injective envelope}} | |||
As stated above, any abelian group ''A'' can be uniquely embedded in a divisible group ''D'' as an [[essential subgroup]]. This divisible group ''D'' is the '''injective envelope''' of ''A'', and this concept is the [[injective hull]] in the category of abelian groups. | |||
==Reduced abelian groups== | |||
An abelian group is said to be '''reduced''' if its only divisible subgroup is {0}. Every abelian group is the direct sum of a divisible subgroup and a reduced subgroup. In fact, there is a unique largest divisible subgroup of any group, and this divisible subgroup is a direct summand.<ref>Griffith, p.7</ref> This is a special feature of [[hereditary ring]]s like the integers '''Z''': the [[direct sum of modules|direct sum]] of injective modules is injective because the ring is [[Noetherian ring|Noetherian]], and the quotients of injectives are injective because the ring is hereditary, so any submodule generated by injective modules is injective. The converse is a result of {{harv|Matlis|1958}}: if every module has a unique maximal injective submodule, then the ring is hereditary. | |||
A complete classification of countable reduced periodic abelian groups is given by [[Ulm's theorem]]. | |||
==Generalization== | |||
Several distinct definitions which generalize divisible groups to divisible modules. The following definitions have been used in the literature to define a divisible [[module (mathematics)|module]] ''M'' over a [[ring (mathematics)|ring]] ''R'': | |||
# ''rM''=''M'' for all nonzero ''r'' in ''R''.{{sfn|Feigelstock|2006}} (It is sometimes required that ''r'' is not a zero-divisor, and some authors{{sfn|Cartan|Eilenberg|1999}}{{sfn|Rotman|2009}} require that ''R'' is a domain.) | |||
# For every principal left ideal ''Ra'', any homomorphism from ''Ra'' into ''M'' extends to a homomorphism from ''R'' into ''M''.{{sfn|Lam|1999}}{{sfn|Nicholson|Yousif | |||
|2003}} (This type of divisible module is also called ''principally injective module''.) | |||
# For every [[finitely generated module|finitely generated]] left ideal ''L'' of ''R'', any homomorphism from ''L'' into ''M'' extends to a homomorphism from ''R'' into ''M''.{{sfn|Damiano|1979}} | |||
The last two conditions are "restricted versions" of the [[Baer's criterion]] for [[injective module]]s. Since injective left modules extend homomorphisms from ''all'' left ideals to ''R'', injective modules are clearly divisible in sense 2 and 3. | |||
If ''R'' is additionally a [[domain (ring theory)|domain]] then all three definitions coincide. If ''R'' is a principal left ideal domain, then divisible modules coincide with injective modules.{{sfn|Lam|1999|loc=p.70—73}} Thus in the case of the ring of integers '''Z''', which is a principal ideal domain, a '''Z'''-module (which is exactly an abelian group) is divisible if and only if it is injective. | |||
If ''R'' is a commutative domain, then the injective ''R'' modules coincide with the divisible ''R'' modules if and only if ''R'' is a [[Dedekind domain]].{{sfn|Lam|1999|loc=p.70—73}} | |||
==Notes== | |||
{{reflist|colwidth=30em}} | |||
==References== | |||
*{{citation |last1=Cartan |first1=Henri |last2=Eilenberg |first2=Samuel |title=Homological algebra |series=Princeton Landmarks in Mathematics |note=With an appendix by David A. Buchsbaum; Reprint of the 1956 original |publisher=Princeton University Press |place=Princeton, NJ |year=1999 |pages=xvi+390 |isbn=0-691-04991-2 |mr=1731415}} | |||
*{{citation |author=Feigelstock, Shalom |title=Divisible is injective |journal=Soochow J. Math. |volume=32 |year=2006 |number=2|pages=241–243 |issn=0250-3255 |mr=2238765}} | |||
* {{cite book | last=Griffith|first=Phillip A. | title=Infinite Abelian group theory | series=Chicago Lectures in Mathematics | publisher=University of Chicago Press | year=1970 | isbn=0-226-30870-7 }} | |||
* {{cite book | title=The theory of groups |last=Hall|first=Marshall, jr | authorlink=Marshall Hall (mathematician) | location=New York | publisher=Macmillan | year=1959 }} Chapter 13.3. | |||
*{{Citation | last1=Lam | first1=Tsit-Yuen | title=Lectures on modules and rings | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Graduate Texts in Mathematics No. 189 | isbn=978-0-387-98428-5 | mr=1653294 | year=1999}} | |||
* {{cite book | title=Algebra, Second Edition | author=Serge Lang | authorlink=Serge Lang | location=Menlo Park, California | publisher=Addison-Wesley | year=1984 }} | |||
*{{Cite journal | last=Matlis|first=Eben | title=Injective modules over Noetherian rings | url=http://projecteuclid.org/getRecord?id=euclid.pjm/1103039896 | mr=0099360 | year=1958 | journal=Pacific Journal of Mathematics | issn=0030-8730 | volume=8 | pages=511–528}} | |||
*{{citation |last1=Nicholson|first1=W. K. |last2=Yousif|first2=M. F. |title=Quasi-Frobenius rings |series=Cambridge Tracts in Mathematics |volume=158 |publisher=Cambridge University Press |place=Cambridge |year=2003 |pages=xviii+307 |isbn=0-521-81593-2 |mr=2003785 |doi=10.1017/CBO9780511546525}} | |||
[[Category:Abelian group theory]] | |||
[[Category:Properties of groups]] | |||
Revision as of 05:07, 4 February 2014
In mathematics, especially in the field of group theory, a divisible group is an abelian group in which every element can, in some sense, be divided by positive integers, or more accurately, every element is an nth multiple for each positive integer n. Divisible groups are important in understanding the structure of abelian groups, especially because they are the injective abelian groups.
Definition
An abelian group G is divisible if, for every positive integer n and every g in G, there exists y in G such that ny = g.[1] An equivalent condition is: for any positive integer n, nG = G, since the existence of y for every n and g implies that nG ⊇ G, and in the other direction nG ⊆ G is true for every group. A third equivalent condition is that an abelian group G is divisible if and only if G is an injective object in the category of abelian groups; for this reason, a divisible group is sometimes called an injective group.
An abelian group is p-divisible for a prime p if for every positive integer n and every g in G, there exists y in G such that pny = g. Equivalently, an abelian group is p-divisible if and only if pG = G.
Examples
- The rational numbers form a divisible group under addition.
- More generally, the underlying additive group of any vector space over is divisible.
- Every quotient of a divisible group is divisible. Thus, is divisible.
- The p-primary component of , which is isomorphic to the p-quasicyclic group is divisible.
- The multiplicative group of the complex numbers is divisible.
- Every existentially closed group (in the model theoretic sense) is divisible.
- The space of orientation-preserving isometries of is divisible. This is because each such isometry is either a translation or a rotation about a point, and in either case the ability to "divide by n" is plainly present. This is the simplest example of a non-Abelian divisible group.
Properties
- If a divisible group is a subgroup of an abelian group then it is a direct summand.[2]
- Every abelian group can be embedded in a divisible group.[3]
- Non-trivial divisible groups are not finitely generated.
- Further, every abelian group can be embedded in a divisible group as an essential subgroup in a unique way.[4]
- An abelian group is divisible if and only if it is p-divisible for every prime p.
- Let be a ring. If is a divisible group, then is injective in the category of -modules.[5]
Structure theorem of divisible groups
Let G be a divisible group. Then the torsion subgroup Tor(G) of G is divisible. Since a divisible group is an injective module, Tor(G) is a direct summand of G. So
As a quotient of a divisible group, G/Tor(G) is divisible. Moreover, it is torsion-free. Thus, it is a vector space over Q and so there exists a set I such that
The structure of the torsion subgroup is harder to determine, but one can showPotter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park. that for all prime numbers p there exists such that
where is the p-primary component of Tor(G).
Thus, if P is the set of prime numbers,
Injective envelope
Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. As stated above, any abelian group A can be uniquely embedded in a divisible group D as an essential subgroup. This divisible group D is the injective envelope of A, and this concept is the injective hull in the category of abelian groups.
Reduced abelian groups
An abelian group is said to be reduced if its only divisible subgroup is {0}. Every abelian group is the direct sum of a divisible subgroup and a reduced subgroup. In fact, there is a unique largest divisible subgroup of any group, and this divisible subgroup is a direct summand.[6] This is a special feature of hereditary rings like the integers Z: the direct sum of injective modules is injective because the ring is Noetherian, and the quotients of injectives are injective because the ring is hereditary, so any submodule generated by injective modules is injective. The converse is a result of Template:Harv: if every module has a unique maximal injective submodule, then the ring is hereditary.
A complete classification of countable reduced periodic abelian groups is given by Ulm's theorem.
Generalization
Several distinct definitions which generalize divisible groups to divisible modules. The following definitions have been used in the literature to define a divisible module M over a ring R:
- rM=M for all nonzero r in R.Template:Sfn (It is sometimes required that r is not a zero-divisor, and some authorsTemplate:SfnTemplate:Sfn require that R is a domain.)
- For every principal left ideal Ra, any homomorphism from Ra into M extends to a homomorphism from R into M.Template:SfnTemplate:Sfn (This type of divisible module is also called principally injective module.)
- For every finitely generated left ideal L of R, any homomorphism from L into M extends to a homomorphism from R into M.Template:Sfn
The last two conditions are "restricted versions" of the Baer's criterion for injective modules. Since injective left modules extend homomorphisms from all left ideals to R, injective modules are clearly divisible in sense 2 and 3.
If R is additionally a domain then all three definitions coincide. If R is a principal left ideal domain, then divisible modules coincide with injective modules.Template:Sfn Thus in the case of the ring of integers Z, which is a principal ideal domain, a Z-module (which is exactly an abelian group) is divisible if and only if it is injective.
If R is a commutative domain, then the injective R modules coincide with the divisible R modules if and only if R is a Dedekind domain.Template:Sfn
Notes
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References
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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.
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The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - 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