View source for Rng (algebra)

{{short description|Algebraic ring without a multiplicative identity}}
{{Algebraic structures |Ring}}
In [[Mathematics|mathematics]], and more specifically in [[Abstract algebra|abstract algebra]], a '''rng''' (or '''non-unital ring''' or '''[[Pseudo-ring|pseudo-ring]]''') is an [[Algebraic structure|algebraic structure]] satisfying the same properties as a [[Ring (mathematics)|ring]], but without assuming the existence of a multiplicative identity. The term ''rng'' (IPA: {{IPAc-en|r|ʌ|ŋ}}) is meant to suggest that it is a ring without ''i'', that is, without the requirement for an identity element.{{sfnp|Jacobson|1989|pp=155–156|ps=}}

There is no consensus in the community as to whether the existence of a multiplicative identity must be one of the ring axioms (see ''{{slink|Ring (mathematics)|History}}''). The term ''rng'' was coined to alleviate this ambiguity when people want to refer explicitly to a ring without the axiom of multiplicative identity.

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.

== Definition ==

Formally, a '''rng''' is a [[Set (mathematics)|set]] ''R'' with two binary operations {{nowrap|(+, ·)}} called ''addition'' and ''multiplication'' such that
* (''R'', +) is an [[Abelian group|abelian group]],
* (''R'', ·) is a [[Semigroup|semigroup]],
* Multiplication distributes over addition.

A '''rng homomorphism''' is a function {{nowrap|''f'': ''R'' → ''S''}} from one rng to another such that
* ''f''(''x'' + ''y'') = ''f''(''x'') + ''f''(''y'')
* ''f''(''x'' · ''y'') = ''f''(''x'') · ''f''(''y'')
for all ''x'' and ''y'' in ''R''.

If ''R'' and ''S'' are rings, then a [[Ring homomorphism|ring homomorphism]] {{nowrap|''R'' → ''S''}} is the same as a rng homomorphism {{nowrap|''R'' → ''S''}} that maps 1 to 1.

== Examples ==

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.

Rngs often appear naturally in [[Functional analysis|functional analysis]] when linear operators on infinite-dimensional [[Vector space|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 composition of operators, this is a rng, but not a ring. Another example is the rng of all real [[Sequence|sequence]]s that [[Limit of a sequence|converge to]] 0, with component-wise operations.

Also, many test function spaces occurring in the theory of distributions consist of functions
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|continuous function]]s with [[Compact space|compact]] [[Support (mathematics)|support]] defined on some [[Topological space|topological space]], together with pointwise addition and multiplication, form a rng; this is not a ring unless the underlying space is [[Compact space|compact]].

=== Example: even integers ===

The set 2'''Z''' of even integers is closed under addition and multiplication and has an additive identity, 0, so it is a rng, but it does not have a multiplicative identity, so it is not a ring.

In 2'''Z''', the only multiplicative [[Idempotence|idempotent]] is 0, the only [[Nilpotent|nilpotent]] is 0, and the only element with a [[Generalized inverse|reflexive inverse]] is 0.

=== Example: finite [[Quinary|quinary]] sequences ===

The direct sum <math display="inline">\mathcal T = \bigoplus_{i=1}^\infty \mathbf{Z}/5 \mathbf{Z}</math> equipped with coordinate-wise addition and multiplication is a rng with the following properties:
* Its [[Idempotence|idempotent]] elements form a lattice with no upper bound.
* Every element ''x'' has a [[Generalized inverse|reflexive inverse]], namely an element ''y'' such that {{nowrap|1=''xyx'' = ''x''}} and {{nowrap|1=''yxy'' = ''y''}}.
* For every finite subset of <math>\mathcal T</math>, there exists an idempotent in <math>\mathcal T</math> that acts as an identity for the entire subset: the sequence with a one at every position where a sequence in the subset has a non-zero element at that position, and zero in every other position.

== Properties ==

{{bulleted list
|Ideals, [[Quotient ring|quotient ring]]s, and modules can be defined for rngs in the same manner as for rings.  
|Working with rngs instead of rings complicates some related definitions, however.  For example, in a ring ''R'', the left ideal (''f'') generated by an element ''f'', defined as the smallest left ideal containing ''f'', is simply ''Rf'', but if ''R'' is only a rng, then ''Rf'' might not contain ''f'', so instead 
<math display="block">(f) = Rf + \mathbf{Z} f = \{ af+nf : a \in R ~\text{and}~ n\in \mathbf{Z} \},</math>
where ''nf'' must be interpreted using repeated addition/subtraction since ''n'' need not represent an element of ''R''.  Similarly, the left ideal generated by elements ''f''{{sub|1}}, ..., ''f''{{sub|''m''}} of a rng ''R'' is
<math display="block">(f_1,\ldots,f_m) = \{ a_1 f_1 + \cdots + a_m f_m + n_1 f_1 + \cdots n_m f_m : a_i \in R \;\mathrm{and}\; n_i\in \mathbf{Z} \},</math> 
a formula that goes back to [[Biography:Emmy Noether|Emmy Noether]].{{sfnp|Noether|1921|loc=p.&nbsp;30, §1.2|ps=}}  Similar complications arise in the definition of submodule generated by a set of elements of a module.
|Some theorems for rings are false for rngs.  For example, in a ring, every proper ideal is contained in a [[Maximal ideal|maximal ideal]], so a nonzero ring always has at least one maximal ideal.  Both these statements fail for rngs.
|A rng homomorphism {{nowrap|''f'' : ''R'' → ''S''}} maps any idempotent element to an idempotent element.
|If {{nowrap|''f'' : ''R'' → ''S''}} is a rng homomorphism from a ring to a rng, and the image of ''f'' contains a non-zero-divisor of ''S'', then ''S'' is a ring, and ''f'' is a ring homomorphism.
}}

== Adjoining an identity element (Dorroh extension) ==

Every rng ''R'' can be enlarged to a ring ''R''^ by adjoining an identity element.  A 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'' with the premise that none of its nonzero integral multiples coincide or are contained in ''R''.  That is, elements of ''R''^ are of the form
{{block indent | em = 1.6 | text = ''n'' &sdot; 1 + ''r''}}
where ''n'' is an [[Integer|integer]] and {{nowrap|''r'' ∈ ''R''}}.  Multiplication is defined by linearity:
{{block indent | em = 1.6 | text = (''n''<sub>1</sub> + ''r''<sub>1</sub>) &sdot; (''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>.}}

More formally, we can take ''R''^ to be the [[Cartesian product|cartesian product]] {{nowrap|'''Z''' × ''R''}} and define addition and multiplication by
{{block indent | em = 1.6 | text = (''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>),}}
{{block indent | em = 1.6 | text = (''n''<sub>1</sub>, ''r''<sub>1</sub>) &middot; (''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|universal property]]:
{{block indent | em = 1.6 | text = 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''}}.}}
The map ''g'' can be defined by {{nowrap|1=''g''(''n'', ''r'') = ''n'' · 1<sub>''S''</sub> + ''f''(''r'')}}.

There is a natural surjective ring homomorphism {{nowrap|''R''^ → '''Z'''}} which sends {{nowrap|(''n'', ''r'')}} to ''n''.  The 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|quotient ring]] ''R''^/''R'' isomorphic to '''Z'''.  It follows that
{{block indent | em = 1.6 | text = ''Every rng is an ideal in some ring, and every ideal of a ring is a rng.''}}

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. The ring ''R''^ is often called the '''Dorroh extension''' of ''R'' after the American mathematician Joe Lee Dorroh, who first constructed it.

The process of adjoining an identity element to a rng can be formulated in the language of [[Category theory|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|subcategory]] of '''Rng'''.  The construction of ''R''^ given above yields a left adjoint to the  inclusion functor {{nowrap|''I'' : '''Ring''' → '''Rng'''}}.  Notice that '''Ring''' is not a [[Reflective subcategory|reflective subcategory]] of '''Rng''' because the inclusion functor is not full.

== Properties weaker than having an identity ==

There are several properties that have been considered in the literature that are weaker than having an identity element, but not so general. 
For example:
* 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 orthogonal (i.e. {{nowrap|1=''ef'' = 0}} for all {{nowrap|''e'' ≠ ''f''}} in ''E'') idempotents (i.e. {{nowrap|1=''e''<sup>2</sup> = ''e''}} for all ''e'' in ''E'') such that {{nowrap|1=''R'' = {{big|⊕}}<sub>''e''∈''E''</sub> ''eR'' = {{big|⊕}}<sub>''e''∈''E''</sub> ''Re''}}.
* 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''.
* ''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''.
* 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.
* 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 display="inline">r = \sum_i r_i s_i</math>.

It is not hard to check that these properties are weaker than having an identity element and weaker than the previous one.
* 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.
* Rings with enough idempotents are rings with local units just taking finite sums of the orthogonal idempotents to satisfy the definition.
* Rings with local units are in particular ''s''-unital; ''s''-unital rings are firm and firm rings are idempotent.

== Rng of square zero ==

A '''rng of square zero''' is a rng ''R'' such that {{nowrap|1=''xy'' = 0}} for all ''x'' and ''y'' in ''R''.{{refn|See {{harvp|Bourbaki|1998|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; see e.g. {{harvp|Szele|1949}} and {{harvp|Kreinovich|1995}}.}}
Any [[Abelian group|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'';{{sfnp|Bourbaki|1998|p=102|ps=}} thus every abelian group is the additive group of some rng.
The only rng of square zero with a multiplicative identity is the [[Zero ring|zero ring]] {0}.{{sfnp|Bourbaki|1998|p=102|ps=}}

Any additive [[Subgroup|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|cyclic group]] of prime order.{{sfnp|Zariski|Samuel|1958|p=133|ps=}}

== Unital homomorphism ==

Given two unital algebras ''A'' and ''B'', an algebra [[Homomorphism|homomorphism]]
{{block indent | em = 1.6 | text = ''f'' : ''A'' → ''B''}}
is '''unital''' if it maps the identity element of ''A'' to the identity element of ''B''.

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|vector space]] and define multiplication ∗ by
{{block indent | em = 1.6 | text = (''x'', ''r'') ∗ (''y'', ''s'') = (''xy'' + ''sx'' + ''ry'', ''rs'')}}
for ''x'', ''y'' in ''A'' and ''r'', ''s'' in ''K''. Then ∗ is an associative operation with identity element {{nowrap|(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 constructions.

== See also ==
* [[Semiring]]

== Citations ==
{{reflist}}

== References ==
{{refbegin}}
* {{cite book
  | title=Algebra I, Chapters 1–3
  | publisher=Springer
  | year=1998
  }}
* {{cite book
  | last1=Dummit | first1=David S.
  | last2=Foote | first2=Richard M.
  | title=Abstract Algebra
  | edition=3rd
  | publisher=Wiley
  | year=2003
  | isbn=978-0-471-43334-7
  }}
* {{cite journal
  | last1 = Dorroh | first1 = J. L.
  | doi = 10.1090/S0002-9904-1932-05333-2
  | journal = Bull. Amer. Math. Soc.
  | volume = 38
  | pages = 85–88
  | title = Concerning Adjunctions to Algebras
  | year = 1932
  | issue = 2
  | doi-access = free
  }}
* {{cite book
  | last1=Jacobson | first1=Nathan
  | title=Basic algebra
  | date=1989
  | publisher=W.H. Freeman | location=New York
  | isbn=0-7167-1480-9
  | edition=2nd
  | url=https://archive.org/details/basicalgebra00jaco
  }}
* {{cite journal
  | last = Kreinovich | first = V.
  | doi = 10.1007/BF01190935
  | issue = 2
  | journal = Algebra Universalis
  | mr = 1318988
  | pages = 237–242
  | title = If a polynomial identity guarantees that every partial order on a ring can be extended, then this identity is true only for a zero-ring
  | volume = 33
  | year = 1995
  | s2cid = 122388143
  }}
* {{cite book
  | last=Herstein | first=I. N.
  | title=Abstract Algebra
  | edition=3rd
  | publisher=Wiley
  | year=1996
  | isbn = 978-0-471-36879-3
  }}
* {{cite book
  | last=McCrimmon | first=Kevin
  | title=A taste of Jordan algebras
  | publisher=Springer
  | year=2004
  | isbn=978-0-387-95447-9
  }}
* {{cite journal
  | date        = 1921
  | title       = Idealtheorie in Ringbereichen
  | trans-title = Ideal theory in rings
  | url         = https://doi.org/10.1007/BF01464225
  | language    = German
  | journal     = Mathematische Annalen
  | volume      = 83
  | issue       = 1–2
  | pages       = 24–66
  | doi         = 10.1007/BF01464225
  | s2cid = 121594471
  | access-date = 
  }}
* {{cite journal
  | last = Szele | first = Tibor
  | year = 1949
  | journal = Mathematische Annalen
  | mr = 0033822
  | pages = 242–246
  | title = Zur Theorie der Zeroringe
  | volume = 121
  | doi=10.1007/bf01329628| s2cid = 122196446
  }}
* {{cite book
  | last1=Zariski | first1=Oscar
  | last2=Samuel | first2=Pierre
  | year=1958
  | title=Commutative Algebra
  | volume=1
  | publisher=Van Nostrand
  }}
{{refend}}

{{DEFAULTSORT:Rng (Algebra)}}
[[Category:Ring theory]]
[[Category:Algebraic structures]]
[[Category:Algebras]]

he:חוג (מבנה אלגברי)#איבר יחידה

{{Sourceattribution|Rng (algebra)}}