symmetric monoidal (∞,1)-category of spectra
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.
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).
Specifically:
A nonunital ring or rng is a set $R$ with operations of addition and multiplication, such that:
More sophisticatedly, we can say that, just as a ring is a monoid object in Ab, so
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.
nonunital Ek-algebras are discussed in (Lurie, section 5.2.3).
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
where for $n \in \mathbb{Z}$ and $a \in A$ we set $n a \coloneqq \underset{n\;summands}{\underbrace{a + a + \cdots + a}}$.
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.
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 $ax - a = 0, xa - a = 0$, so that this $x$ must be a right and left identity for multiplication $a$. Cosets must be represented by expressions of the form $a + nx$; this provides an obvious isomorphism to the above definition, and motivates the multiplication for $\mathbb{Z} \oplus A$ above.
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.
Similar unitisation prescriptions work for non-commutative rings and for nonunital algebras over a fixed base ring, see also at
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.
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.
Write $CRing_{/\mathbb{Z}}$ for the slice category of CRing over the ring of integers (augmented commutative rings). Write
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 $\phi$
The augmentation ideal-functor in def. is an equivalence of categories whose inverse is given by unitisation, def. , remembering the projection $(\mathbb{Z} \oplus A) \to \mathbb{Z}$:
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 $\mathbb{Z}$
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).
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).
The generalization of prop. to nonunital Ek-algebras is (Lurie, prop. 5.2.3.15).
A survey of various notions between unital rings and nonunital rings:
The terminology “rng” originates in
A survey of commutative rng theory is in
Discussion of module theory over rngs (with close relation to torsion modules) is in
Discussion of non-unital rings as augmentation ideals of augmented unital rings includes
A definition of algebraic K-theory for nonunital rings is due to
with further developments (in KK-theory) including
Discussion in the context of higher algebra (nonunital Ek-algebras) is in
Last revised on October 4, 2023 at 12:57:49. See the history of this page for a list of all contributions to it.