Rng (algebra)

Algebraic structures
Group-like Group Semigroup / Monoid Rack and quandle Quasigroup and loop Abelian group Magma Lie group Group theory
Ring-like Ring Rng Semiring Near-ring Commutative ring Domain Integral domain Field Division ring Lie ring Ring theory
Lattice-like Lattice Semilattice Complemented lattice Total order Heyting algebra Boolean algebra Map of lattices Lattice theory
Module-like Module Group with operators Vector space Linear algebra
Algebra-like Algebra Associative Non-associative Composition algebra Lie algebra Graded Bialgebra Hopf algebra
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In

There is no consensus in the community as to whether the existence of a multiplicative identity must be one of the

). 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

) space.

Definition

Formally, a rng is a

binary operations

(+, ·) called addition and multiplication such that

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

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.

If R and S are rings, then a ring homomorphism R → S is the same as a rng homomorphism 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

is a rng.

Rngs often appear naturally in

0, with component-wise operations.

Also, many

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.

Example: even integers

The set 2Z 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 2Z, the only multiplicative idempotent is 0, the only nilpotent is 0, and the only element with a reflexive inverse is 0.

Example: finite quinary sequences

The direct sum ${\textstyle {\mathcal {T}}=\bigoplus _{i=1}^{\infty }\mathbf {Z} /5\mathbf {Z} }$ equipped with coordinate-wise addition and multiplication is a rng with the following properties:

• Its idempotent elements form a lattice with no upper bound.
• Every element x has a reflexive inverse, namely an element y such that xyx = x and yxy = y.
• For every finite subset of ${\displaystyle {\mathcal {T}}}$, there exists an idempotent in ${\displaystyle {\mathcal {T}}}$ 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

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

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

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

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:

The map g can be defined by g(n, r) = n · 1_S + f(r).

There is a natural

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 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

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. ef = 0 for all e ≠ f in E) idempotents (i.e. e² = e for all e 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 ${\textstyle r=\sum _{i}r_{i}s_{i}}$.

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.[3] Any abelian group can be made a rng of square zero by defining the multiplication so that xy = 0 for all x and y;^[4] 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}.^[4]

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.^[5]

Unital homomorphism

Given two unital algebras A and B, an algebra homomorphism

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

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

.

See also

• Semiring

Citations

References