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rng

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Idea

The concept of nonunital ring is like that of ring but without the requirement of the existence of an identity element (“unit” element).

Historically, this was in fact the original meaning of “ring”, and while mostly “ring” has come to mean by default the version with identity element, nonunital rings still play a role (see e.g. the review in Anderson 06) and in some areas of mathematics “nonunital ring” is still the default meaning of “ring”. In particular, non-unital rings may naturally be identified with \mathbb{Z}-augmented unital rings, see the discussion below.

Remark on terminology

The term “non-unital ring” may be regarded as an example of the “red herring principle”, as a non-unital ring is not a ring in the modern sense of the word.

In Bourbaki 6, chapter 1 the term pseudo-ring is used, but that convention has not become established.

Another terminology that has been suggested for “nonunital ring”, and which is in use in part of the literature (e.g. Anderson 06) is “rng”, where dropping the “i” in “ring” is meant to be alluding to the absence of identity elements. This terminology appears in print first in (Jacobson), where it is attributed to Louis Rowen. Similarly there is, for whatever it’s worth, the suggestion that a ring without negatuves, hence a semiring, should be called a rig.

Definitions

Specifically:

Definition

A nonunital ring or rng is a set RR with operations of addition and multiplication, such that:

  • RR is a semigroup under multiplication;
  • RR is an abelian group under addition;
  • multiplication distributes over addition.

More sophisticatedly, we can say that, just as a ring is a monoid object in Ab, so

Definition

A nonuntial ring or rng is a semigroup object in Ab.

Properties

Unitization

Definition

Given a non-unital commutative ring AA, then its unitisation is the commutative ring F(A)F(A) obtained by freely adjoining an identity element, hence the ring whose underlying abelian group is the direct sum A\mathbb{Z} \oplus A of AA with the integers, and whose product operation is defined by

(n 1,a 1)(n 2,a 2)(n 1n 2,n 1a 2+n 2a 1+a 1a 2), (n_1, a_1) (n_2, a_2) \coloneqq (n_1 n_2, n_1 a_2 + n_2 a_1 + a_1 a_2) \,,

where for nn \in \mathbb{Z} and aAa \in A we set naa+a++ansummandsn a \coloneqq \underset{n\;summands}{\underbrace{a + a + \cdots + a}}.

Remark

In the unitization A\mathbb{Z} \oplus A we have (n,0)+(0,a)=(n,a)(n,0) + (0,a) = (n,a) and hence it makes sense to abbreviate (n,a)(n,a) simply to n+an+a. The product in the unitisation is then fixed by the defining requirement that 1a=a1 \cdot a = a and by the distributivity law.

Remark

Similar unitisation prescriptions work for non-commutative rings and for associative algebras over a fixed base ring, see also at

Proposition

Unitisation in def. 3 extends to a functor from CRngCRng to CRing which is left adjoint to the forgetful functor from commutative rings to non-unital commutative rings.

F:CRngCRing:U. F \colon CRng \leftrightarrow CRing \colon U \,.
Proof

This is because the definition of any ring homomorphism out of F(A)=(A,)F(A)= (\mathbb{Z} \oplus A, \cdot) is uniquely fixed on the \mathbb{Z}-summand.

Nonuntial rings as slices of rings

Definition

Write CRing /CRing_{/\mathbb{Z}} for the slice category of CRing over the ring of integers (augmented commutative rings). Write

CRing /CRng CRing_{/\mathbb{Z}} \longrightarrow CRng

for the functor to commutative non-unital rings which sends any (Rϕ)(R \stackrel{\phi}{\to} \mathbb{Z}) to its augmentation ideal, hence to the kernel of ϕ\phi

(Rϕ)ker(ϕ). (R \stackrel{\phi}{\to} \mathbb{Z}) \mapsto ker(\phi) \,.
Proposition

The augmentation ideal-functor in def. 5 is an equivalence of categories whose inverse is given by unitisation, def. 3, remembering the projection (A)(\mathbb{Z} \oplus A) \to \mathbb{Z}:

CRngCRing /. CRng \simeq CRing_{/\mathbb{Z}} \,.
Proof

That the functor is fully faithful is to observe that for a ring RϕR \stackrel{\phi}{\to} \mathbb{Z} the fiber R nR_{n} over nn \in \mathbb{Z} is a torsor over the additive group underlying the augmentation ideal A=R 0=ker(ϕ)A = R_0 = ker(\phi), and moreover it is a pointed torsor, the point being nn itself, hence is canonically equivalent to the augmentation ideal AA, the equivalence being addition by nn in RR. Hence any homomrphism of rings with identity over \mathbb{Z}

R 1 f R 2 ϕ 1 ϕ 2 \array{ R_1 && \stackrel{f}{\longrightarrow} && R_2 \\ & {}_{\mathllap{\phi_1}}\searrow && \swarrow_{\mathrlap{\phi_2}} \\ && \mathbb{Z} }

is uniquely fixed by its restriction to the augmentation ideal ker(ϕ 1)ker(\phi_1), whose image, moreover, has to be in the augmentation ideal ker(ϕ 2)ker(\phi_2).

Remark

In terms of arithmetic geometry, the formally dual statement of prop. 5 is that arithmetic geometry induced by non-unital rings is equivalently ordinary arithmetic geometry under Spec(Z).

References

Nonunital ring theory

A survey of commutative rng theory is in

  • D. Anderson, Commutative rngs, in J. Brewer et al. (eds.) Multiplicative ideal theory in Commutative Algebra, 2006

Discussion of module theory over rngs is in

Terminology

The notation “rng” originates in

  • Nathan Jacobson Basic Algebra,

where the term is attributed to Louis Rowen.

(Bourbaki 6, chapter 1) uses the term “pseudo-ring” instead, which however has not caught on and even if more sane, will be understood less than “rng”.

Revised on August 20, 2014 06:35:56 by Urs Schreiber (89.204.139.117)