Rng (algebra)

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, 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 the requirement for an "identity element".

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 history section of the article on 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.

A number of algebras of functions considered in 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.

Definition

Formally, a rng is a set R with two binary operations (+, ·) called addition and multiplication such that

(R, +) is an abelian group,
(R, ·) is a semigroup,
Multiplication distributes over addition.

Rng homomorphisms are defined in the same way as ring homomorphisms except that the requirement f(1) = 1 is dropped. That is, a rng homomorphism is a function 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.

Examples

All rings are rngs. A simple example of a rng that is not a ring is given by the even integers with the ordinary addition and multiplication of integers. Another example is given by the set of all 3-by-3 real matrices whose bottom row is zero. Both of these examples are instances of the general fact that every (one- or two-sided) ideal is a rng.

Rngs often appear naturally in functional analysis when linear operators on infinite-dimensional vector spaces are considered. Take for instance any infinite-dimensional vector space V and consider the set of all linear operators f : V → V with finite rank (i.e. 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 sequences that 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 functions with compact 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.

Properties

Ideals and quotient rings 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 ideals. Some theorems of ring theory are false for rngs.

A rng homomorphism f: R → S maps any idempotent element to an idempotent element; this applies in particular to 1_R if it exists.

If R and S are rings, a rng homomorphism f: R → S whose image contains a non-zero-divisor maps 1_R to 1_S.

Adjoining an identity element

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

n · 1 + r

where n is an integer and r ∈ R. Multiplication is defined by linearity:

(n₁ + r₁) · (n₂ + r₂) = n₁n₂ + n₁r₂ + n₂r₁ + r₁r₂.

More formally, we can take R^ to be the cartesian product Z × R and define addition and multiplication by

(n₁, r₁) + (n₂, r₂) = (n₁ + n₂, r₁ + r₂),

(n₁, r₁) · (n₂, r₂) = (n₁n₂, n₁r₂ + n₂r₁ + r₁r₂).

The multiplicative identity of R^ is then (1, 0). There is a natural rng homomorphism j : R → R^ defined by j(r) = (0, r). This map has the following universal property:

Given any ring S and any rng homomorphism f : R → S, there exists a unique ring homomorphism g : R^ → S such that f = gj.

The map g can be defined by g(n, r) = n · 1_S + f(r). In a sense then, R^ is "the most general" ring containing R.

There is a natural surjective ring homomorphism R^ → Z which sends (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 in R^ with the quotient ring R^/R isomorphic to Z. It follows that

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 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 I : Ring → Rng. This means that Ring is a reflective subcategory of Rng with reflector j : R → R^.

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 idempotents (i.e. e² = e for all e in E) orthogonal (i.e. ef = 0 for all e ≠ f in E) such that R = ⊕_e∈E eR = ⊕_e∈E Re.
Rings with local units: A rng R is said to be a ring with local units in case for every finite set r₁, r₂, ..., r_t in R we can find e in R such that e² = e and er_i = r_i = r_ie for every i.
s-unital rings: A rng R is said to be s-unital in case for every finite set r₁, r₂, ..., r_t in R we can find s in R such that sr_i = r_i = r_is for every i.
Firm rings: A rng R is said to be firm if the canonical homomorphism R ⊗_R R → R given by r ⊗ s ↦ rs is an isomorphism.
Idempotent rings: A rng R is said to be idempotent (or an irng) in case R² = R, that is, for every element r of R we can find elements r_i and s_i in R such that $r = \Sigma_i r_i s_i$ .

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 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 xy = 0 for all x and y in R.^[1] Any abelian group can be made a rng of square zero by defining the multiplication so that xy = 0 for all x and y;^[2] 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 {0}.^[3]

Any additive subgroup of a rng of square zero is an ideal. Thus a rng of square zero is simple if and only if its additive group is a simple abelian group, i.e., a cyclic group of prime order.^[4]

Unital homomorphism

Given two unital algebras A and B, an algebra homomorphism

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 K is not unital, one can adjoin an identity element as follows: take A × K as underlying K-vector space and define multiplication ∗ by

(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 (0,1). The old algebra A is contained in the new one, and in fact A × K is the "most general" unital algebra containing A, in the sense of universal constructions.

Notes

↑ 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; see e.g. Szele (1949) and Kreinovich (1995).
↑ Bourbaki, p. 102.
↑ Bourbaki, p. 102.
↑ Zariski and Samuel, p. 133.

References

Bourbaki, N. (1998). Algebra I, Chapters 1–3. Springer.
Dummit, David S.; Foote, Richard M. (2003). Abstract Algebra (3rd ed.). Wiley. ISBN 978-0-471-43334-7.
Kreinovich, V. (1995). "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". Algebra Universalis 33 (2): 237–242. doi:10.1007/BF01190935. MR 1318988.
Herstein, I. N. (1996). Abstract Algebra (3rd ed.). Wiley. ISBN 978-0-471-36879-3.
McCrimmon, Kevin (2004). A taste of Jordan algebras. Springer. ISBN 978-0-387-95447-9.
Szele, Tibor (1949). "Zur Theorie der Zeroringe". Mathematische Annalen 121: 242–246. doi:10.1007/bf01329628. MR 0033822.
Zariski, Oscar; Samuel, Pierre (1958). Commutative Algebra 1. Van Nostrand.