nLab nonunital ring

Idea

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

Nonunital rings with homomorphisms between them form the category Rng.

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 the augmentation ideals of $\mathbb{Z}$-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 in general 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 negatives, hence a semiring, should be called a rig (although here one may make a technical distinction about the additive identity).

Definitions

For rings

Specifically:

Definition

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

• $R$ is a semigroup under multiplication;
• $R$ 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 nonunital ring or rng is a semigroup object in Ab.

Remark

A non-unital ring may well contain an element that behaves as the identity element, i.e. it may be in the image of the forgetful functor from unital rings to nonunital rings. But if so, then this element is still not part of the defining data and in particular a homomorphism of non-unital rings need not to preserve whatever identity elements may happen to be present.

For $\mathbb{E}_k$-algebras

nonunital Ek-algebras are discussed in (Lurie, section 5.2.3).

Properties

Unitization

Definition

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

$$(n_1, a_1) (n_2, a_2) \coloneqq (n_1 n_2, n_1 a_2 + n_2 a_1 + a_1 a_2) \,,$$

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

Remark

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

Remark

One can also think of the unitisation as the quotient of the polynomial ring “$A[x]$” ($A$ “with a generic element adjoined”, i.e. a term algebra) quotiented by the relations $a x - a = 0, x a - a = 0$, so that this $x$ must be a right and left identity for multiplication $a$. Cosets must be represented by expressions of the form $a + n x$; this provides an obvious isomorphism to the above definition, and motivates the multiplication for $\mathbb{Z} \oplus A$ above.

Remark

Since $A$ embeds into its unitisation $F(A)$, every rng lives as an ideal in some ring. We can consider $A$-linear actions $A \curvearrowright M$ on abelian groups $M$. Since the endomorphism rings of abelian groups are always unital, the universal property of the unitisation-forgetful adjunction (see below) ensures that there is a unique extension of the action map $A \to \operatorname{End}_{\mathbf{Ab}}(M)$ along $A \hookrightarrow \operatorname{for} \circ F(A)$ to an action map $F(A) \to \operatorname{End}_{\mathbf{Ab}}(M)$. This induces an equivalence $A\text{-}\mathbf{Mod} \simeq F(A)\text{-}\mathbf{Mod},$ so that one may as well study $F(A)$ if one wanted to study $A$ through its module category.

Remark

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

• unitisation of C*-algebras.

Proposition

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

$$F \colon CRng \leftrightarrow CRing \colon U \,.$$

Proof

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

Nonunital rings as slices of rings

Definition

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

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

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

$$(R \stackrel{\phi}{\to} \mathbb{Z}) \mapsto ker(\phi) \,.$$

Proposition

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

$$CRng \simeq CRing_{/\mathbb{Z}} \,.$$

Proof

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

$$\array{ R_1 && \stackrel{f}{\longrightarrow} && R_2 \\ & {}_{\mathllap{\phi_1}}\searrow && \swarrow_{\mathrlap{\phi_2}} \\ && \mathbb{Z} }$$

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

The identification of non-unital algebras as augmentation ideals of augmented unital algebras is used for instance in (Fresse 06).

Remark

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

Remark

The generalization of prop. to nonunital Ek-algebras is (Lurie, prop. 5.2.3.15).

Related concepts

• unitisation of C*-algebras

• noncommutative ring, nonassociative ring

• rig

• nonunital Ek-algebra

• paraunital ring

References

Nonunital ring theory

A survey of various notions between unital rings and nonunital rings:

• Patrik Nystedt, A survey of s-unital and locally unital rings, Revista Integración, temas de matemáticas Escuela de Matemáticas, Universidad Industrial de Santander, Vol. 37, N◦ 2, 2019, pág. 251–260. doi, arXiv.

The terminology “rng” originates in

• Nathan Jacobson Basic Algebra.

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 (with close relation to torsion modules) is in

• Daniel Quillen, Module theory over nonunital rings, August 1996 (pdf)

Discussion of non-unital rings as augmentation ideals of augmented unital rings includes

• Benoit Fresse, The Bar Complex of an E-infinity Algebra, Adv. Math. 223 (2010), pages 2049-2096 (arXiv:math/0601085)

A definition of algebraic K-theory for nonunital rings is due to

• Daniel Quillen, $K_0$ for nonunital rings and Morita invariance, J. Reine Angew. Math., 472:197-217, 1996.

with further developments (in KK-theory) including

• Snigdhayan Mahanta, Higher nonunital Quillen K’-theory, KK-dualities and applications to topological T-dualities, J. Geom. Phys., 61 (5), 875-889, 2011. (pdf)

Discussion in the context of higher algebra (nonunital Ek-algebras) is in

• Jacob Lurie, section 5.2.3 of Higher Algebra