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| In [[mathematics]], and more specifically in [[abstract algebra]], a '''rng''' (or '''[[pseudo-ring]]''' or '''non-unital ring''') is an [[algebraic structure]] satisfying the same properties as a [[ring (mathematics)|ring]], without assuming the existence of a multiplicative identity. The term "rng" (pronounced ''rung'') is meant to suggest that it is a "ring" without "i", i.e. without an "identity element".
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| There is no consensus in the community as to whether the existence of a multiplicative identity must be one of the ring axioms (see the [[Ring (mathematics)#History|history section]] of the article on [[Ring (mathematics)|rings]]). The term "rng" was coined to alleviate this ambiguity when people want to refer explicitly to a ring without the axiom of multiplicative identity.
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| A number of algebras of functions considered in [[Mathematical analysis|analysis]] are not unital, for instance the algebra of functions decreasing to zero at infinity, especially those with [[compact support]] on some (non-[[compact space|compact]]) space.
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| == Definition ==
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| Formally, a '''rng''' is a [[Set (mathematics)|set]] ''R'' with two [[binary operations]] {{nowrap|(+, ·)}} called ''addition'' and ''multiplication'' such that
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| * (''R'', +) is an [[abelian group]],
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| * (''R'', ·) is a [[semigroup]],
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| * Multiplication [[distributive law|distributes]] over addition.
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| Rng [[homomorphism]]s are defined in the same way as [[ring homomorphism]]s except that the requirement {{nowrap|1=''f''(1) = 1}} is dropped. That is, a '''rng homomorphism''' is a function {{nowrap|''f'': ''R'' → ''S''}} from one rng to another such that
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| * ''f''(''x'' + ''y'') = ''f''(''x'') + ''f''(''y'')
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| * ''f''(''x'' · ''y'') = ''f''(''x'') · ''f''(''y'')
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| for all ''x'' and ''y'' in ''R''.
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| == Examples ==
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| All rings are rngs. A simple example of a rng that is not a ring is given by the [[even number|even integers]] with the ordinary addition and multiplication of integers. Another example is given by the set of all 3-by-3 real [[matrix (mathematics)|matrices]] whose bottom row is zero. Both of these examples are instances of the general fact that every (one- or two-sided) [[ideal (ring theory)|ideal]] is a rng.
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| Rngs often appear naturally in [[functional analysis]] when [[linear operator]]s on infinite-[[dimension (linear algebra)|dimensional]] [[vector space]]s are considered. Take for instance any infinite-dimensional vector space ''V'' and consider the set of all linear operators {{nowrap|''f'' : ''V'' → ''V''}} with finite [[rank (linear algebra)|rank]] (i.e. {{nowrap|dim ''f''(''V'') < ∞}}). Together with addition and [[functional composition|composition]] of operators, this is a rng, but not a ring. Another example is the rng of all real [[sequence]]s that [[limit of a sequence|converge to]] 0, with component-wise operations.
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| Also, many [[test function]] spaces occurring in the [[theory of distributions]] consist of functions
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| decreasing to zero at infinity, like e.g. [[Schwartz space]]. Thus, the function everywhere equal to one, which would be the only possible identity element for pointwise multiplication, cannot exist in such spaces, which therefore are rngs (for pointwise addition and multiplication). In particular, the real-valued [[continuous function]]s with [[compact space|compact]] [[support (mathematics)|support]] defined on some [[topological space]], together with pointwise addition and multiplication, form a rng; this is not a ring unless the underlying space is [[compact space|compact]].
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| == Properties ==
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| Ideals and [[quotient ring]]s can be defined for rngs in the same manner as for rings. The ideal theory of rngs is complicated by the fact that a nonzero rng, unlike a nonzero ring, need not contain any [[maximal ideal]]s. Some theorems of [[ring theory]] are false for rngs.
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| A rng homomorphism {{nowrap|''f'': ''R'' → ''S''}} maps any [[idempotent element]] to an idempotent element; this applies in particular to 1<sub>''R''</sub> if it exists.
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| If ''R'' and ''S'' are rings, a rng homomorphism {{nowrap|''f'': ''R'' → ''S''}} whose image contains a non-zero-divisor maps 1<sub>''R''</sub> to 1<sub>''S''</sub>.
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| == Adjoining an identity element ==
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| Every rng ''R'' can be turned into a ring ''R''^ by adjoining an identity element. The most general way in which to do this is to formally add an identity element 1 and let ''R''^ consist of integral linear combinations of 1 and elements of ''R''. That is, elements of ''R''^ are of the form
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| :''n'' · 1 + ''r''
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| where ''n'' is an [[integer]] and {{nowrap|''r'' ∈ ''R''}}. Multiplication is defined by linearity:
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| :(''n''<sub>1</sub> + ''r''<sub>1</sub>) · (''n''<sub>2</sub> + ''r''<sub>2</sub>) = ''n''<sub>1</sub>''n''<sub>2</sub> + ''n''<sub>1</sub>''r''<sub>2</sub> + ''n''<sub>2</sub>''r''<sub>1</sub> + ''r''<sub>1</sub>''r''<sub>2</sub>.
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| More formally, we can take ''R''^ to be the [[cartesian product]] {{nowrap|'''Z''' × ''R''}} and define addition and multiplication by
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| :(''n''<sub>1</sub>, ''r''<sub>1</sub>) + (''n''<sub>2</sub>, ''r''<sub>2</sub>) = (''n''<sub>1</sub> + ''n''<sub>2</sub>, ''r''<sub>1</sub> + ''r''<sub>2</sub>),
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| :(''n''<sub>1</sub>, ''r''<sub>1</sub>) · (''n''<sub>2</sub>, ''r''<sub>2</sub>) = (''n''<sub>1</sub>''n''<sub>2</sub>, ''n''<sub>1</sub>''r''<sub>2</sub> + ''n''<sub>2</sub>''r''<sub>1</sub> + ''r''<sub>1</sub>''r''<sub>2</sub>). | |
| The multiplicative identity of ''R''^ is then {{nowrap|(1, 0)}}. There is a natural rng homomorphism {{nowrap|''j'' : ''R'' → ''R''^}} defined by {{nowrap|1=''j''(''r'') = (0, ''r'')}}. This map has the following [[universal property]]:
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| :Given any ring ''S'' and any rng homomorphism {{nowrap|''f'' : ''R'' → ''S''}}, there exists a unique ring homomorphism {{nowrap|''g'' : ''R''^ → ''S''}} such that {{nowrap|1=''f'' = ''gj''}}.
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| The map ''g'' can be defined by {{nowrap|1=''g''(''n'', ''r'') = ''n'' · 1<sub>''S''</sub> + ''f''(''r'')}}. In a sense then, ''R''^ is "the most general" ring containing ''R''.
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| There is a natural surjective ring homomorphism {{nowrap|''R''^ → '''Z'''}} which sends {{nowrap|(''n'', ''r'')}} to ''n''. The [[kernel (ring theory)|kernel]] of this homomorphism is the image of ''R'' in ''R''^. Since ''j'' is injective, we see that ''R'' is embedded as a (two-sided) [[ideal (ring theory)|ideal]] in ''R''^ with the [[quotient ring]] ''R''^/''R'' isomorphic to '''Z'''. It follows that
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| :''Every rng is an ideal in some ring, and every ideal of a ring is a rng.''
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| Note that ''j'' is never surjective. So even when ''R'' already has an identity element the ring ''R''^ will be a larger one with a different identity.
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| The process of adjoining an identity element to a rng can be formulated in the language of [[category theory]]. If we denote the [[category of all rings]] and ring homomorphisms by '''Ring''' and the category of all rngs and rng homomorphisms by '''Rng''', then '''Ring''' is a (nonfull) [[subcategory]] of '''Rng'''. The construction of ''R''^ given above yields a [[left adjoint]] to the [[inclusion functor]] {{nowrap|''I'' : '''Ring''' → '''Rng'''}}. This means that '''Ring''' is a [[reflective subcategory]] of '''Rng''' with reflector {{nowrap|''j'' : ''R'' → ''R''^}}.
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| == Properties weaker than having an identity ==
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| There are several properties that have been considered in the literature that are weaker than having an identity element, but not so general.
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| For example:
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| * Rings with enough idempotents: A rng ''R'' is said to be a ring with enough idempotents when there exists a subset ''E'' of ''R'' given by idempotents (i.e. {{nowrap|1=''e''<sup>2</sup> = ''e''}} for all ''e'' in ''E'') orthogonal (i.e. {{nowrap|1=''ef'' = 0}} for all {{nowrap|''e'' ≠ ''f''}} in ''E'') such that {{nowrap|1=''R'' = {{big|⊕}}<sub>''e''∈''E''</sub> ''eR'' = {{big|⊕}}<sub>''e''∈''E''</sub> ''Re''}}.
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| * Rings with local units: A rng ''R'' is said to be a ring with local units in case for every finite set ''r''<sub>1</sub>, ''r''<sub>2</sub>, ..., ''r<sub>t</sub>'' in ''R'' we can find ''e'' in ''R'' such that {{nowrap|1=''e''<sup>2</sup> = ''e''}} and {{nowrap|1=''er<sub>i</sub>'' = ''r<sub>i</sub>'' = ''r<sub>i</sub>e''}} for every ''i''.
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| * ''s''-unital rings: A rng ''R'' is said to be ''s''-unital in case for every finite set ''r''<sub>1</sub>, ''r''<sub>2</sub>, ..., ''r<sub>t</sub>'' in ''R'' we can find ''s'' in ''R'' such that {{nowrap|1=''sr<sub>i</sub>'' = ''r<sub>i</sub>'' = ''r<sub>i</sub>s''}} for every ''i''.
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| * Firm rings: A rng ''R'' is said to be firm if the canonical homomorphism {{nowrap|''R'' ⊗<sub>''R''</sub> ''R'' → ''R''}} given by {{nowrap|''r'' ⊗ ''s'' ↦ ''rs''}} is an isomorphism.
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| * Idempotent rings: A rng ''R'' is said to be idempotent (or an irng) in case {{nowrap|1=''R''<sup>2</sup> = ''R''}}, that is, for every element ''r'' of ''R'' we can find elements ''r<sub>i</sub>'' and ''s<sub>i</sub>'' in ''R'' such that <math>r = \Sigma_i r_i s_i</math>.
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| It is not hard to check that these properties are weaker than having an identity element and weaker than the previous one.
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| * Rings are rings with enough idempotents, using {{nowrap|1=''E'' = {1}.}} A ring with enough idempotents that has no identity is for example the ring of infinite matrices over a field with just a finite number of nonzero entries. The matrices that have just 1 over one element in the main diagonal and 0 otherwise are the orthogonal idempotents.
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| * Rings with enough idempotents are rings with local units just taking finite sums of the orthogonal idempotents to satisfy the definition.
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| * Rings with local units are in particular ''s''-unital; ''s''-unital rings are firm and firm rings are idempotent.
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| == Rng of square zero ==
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| A '''rng of square zero''' is a rng ''R'' such that {{nowrap|1=''xy'' = 0}} for all ''x'' and ''y'' in ''R''.<ref>See Bourbaki, p. 102, where it is called a pseudo-ring of square zero. Some other authors use the term "zero ring" to refer to any rng of square zero.{{fact|date=November 2013}}</ref>
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| Any [[abelian group]] can be made a rng of square zero by defining the multiplication so that {{nowrap|1=''xy'' = 0}} for all ''x'' and ''y'';<ref>Bourbaki, p. 102.</ref> thus every abelian group is the additive group of some rng.
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| The only rng of square zero with a multiplicative identity is the [[zero ring]] {0}.<ref>Bourbaki, p. 102.</ref>
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| Any additive [[subgroup]] of a rng of square zero is an [[ideal (ring theory)|ideal]]. Thus a rng of square zero is [[simple ring|simple]] if and only if its additive group is a simple abelian group, i.e., a [[cyclic group]] of prime order.<ref>Zariski and Samuel, p. 133.</ref>
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| == Unital [[homomorphism]] ==
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| Given two unital algebras ''A'' and ''B'', an algebra [[homomorphism]]
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| :''f'' : ''A'' → ''B''
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| is '''unital''' if it maps the identity element of ''A'' to the identity element of ''B''.
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| If the associative algebra ''A'' over the [[field (mathematics)|field]] ''K'' is ''not'' unital, one can adjoin an identity element as follows: take {{nowrap|''A'' × ''K''}} as underlying ''K''-[[vector space]] and define multiplication ∗ by
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| :(''x'',''r'') ∗ (''y'',''s'') = (''xy'' + ''sx'' + ''ry'', ''rs'')
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| for ''x'',''y'' in ''A'' and ''r'',''s'' in ''K''. Then ∗ is an associative operation with identity element (0,1). The old algebra ''A'' is contained in the new one, and in fact {{nowrap|1=''A'' × ''K''}} is the "most general" unital algebra containing ''A'', in the sense of [[universal construction]]s.
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| ==Notes==
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| {{reflist}}
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| == References ==
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| <references />
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| * {{Cite book
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| | last1=Bourbaki
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| | first1=N.
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| | author1-link=Nicolas Bourbaki
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| | title=Algebra I, Chapters 1–3
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| | publisher=Springer
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| | year=1998
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| }}
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| * {{Cite book
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| | last1=Dummit
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| | first1=David S.
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| | last2=Foote
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| | first2=Richard M.
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| | title=Abstract Algebra
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| | edition=3rd
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| | publisher=Wiley
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| | year=2003
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| | isbn=978-0-471-43334-7
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| }}
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| * {{Cite book
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| | last=Herstein
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| | first=I. N.
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| | author-link=Israel Nathan Herstein
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| | title=Abstract Algebra
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| | edition=3rd
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| | publisher=Wiley
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| | year=1996
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| | isbn = 978-0-471-36879-3
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| }}
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| * {{Cite book
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| | last=McCrimmon
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| | first=Kevin
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| | title=A taste of Jordan algebras
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| | publisher=Springer
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| | year=2004
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| | isbn=978-0-387-95447-9
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| }}
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| * {{Cite book
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| | last1=Zariski
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| | first1=Oscar
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| | last2=Samuel
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| | first2=Pierre
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| | title=Commutative Algebra
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| | volume=1
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| | publisher=Van Nostrand
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| | year=1958
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| }}
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| ==See also==
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| * [[Ring (mathematics)]]
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| * [[Ring theory]]
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| * [[Semiring]]
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| {{DEFAULTSORT:Rng (Algebra)}}
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| [[Category:Ring theory]]
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| [[Category:Algebraic structures]]
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| [[Category:Algebras]]
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