Non-unital rings: a few examples

Every ring I've ever heard of is unital, i. e., contains a (unique) element $a$ such that $xa = ax = x$ for every $x$ in it. However, some rings do not have such an element. What are they?

P. S.: one will notice I assumed commutativity. So, for an easier related request, some examples of non-commutative rings would also be appreciated.

Solution 1:

One general source of such examples is functional analysis.

One of the easiest examples to describe is the space $C_{0}{(X)}$ of functions vanishing at infinity, where $X$ is locally compact, with pointwise addition and multiplication as operations. This ring is commutative and it is unital if and only if $X$ is compact.

Another class of examples is formed by the convolution algebra $L^{1}(G)$ of a locally compact group $G$. It is unital if and only if $G$ is discrete and it is commutative if and only if $G$ is commutative. So probably the easiest examples of this kind would be $L^{1}(\mathbb{R})$ or $L^{1}(\mathbb{S^1})$. Related but a bit more complicated are the group $C^{\ast}$-algebras.

A completely different kind of (non-commutative) example would be the algebra of compact operators of an infinite-dimensional Banach space.

Solution 2:

Any ideal in a ring is itself a ring (but generally without unity, unless it's the full ring). So, there are plenty of examples.

Solution 3:

Non-unital rings are employed heavily in the general study of radical theories for rings. Perhaps you will find the following remarks of interest, excerpted from the preface of Gardner and Wiegandt: Radical Theory of Rings, 2004.

Some authors deal exclusively with rings with unity element. This assumption is all right and not restrictive, if the ring is fixed, as in module theory or group ring theory or sometimes investigating polynomial rings and power series rings (if the ring of coefficients does not possess a unity element. the indeterminate x is not a member of the polynomial ring). Dealing, however, simultaneously with several objects in a category of rings, demanding the existence of a unity element leads to a bizarre situation. Rings with unity element include among their fundamental operations the nullary operation $\mapsto$ 1 assigning the unity element. Thus in the category of rings with unity element the morphisms, in particular the monomorphisms, have to preserve also this nullary operation: subrings (i.e. subobjects) have to contain the same unity element, and so a proper ideal with unity element is not a subring, although a ring and a direct summand; there are no infinite direct sums, no nil rings, no Jacobson radical rings, the finite valued linear transformations of an infinite dimensional vector space do not form a ring, etc. Thus, in many, maybe most, branches of ring theory the requirement of the existence of a unity element is not sensible, and therefore unacceptable. This applies also to radical theory. and so in this book rings need not have a unity element.

Solution 4:

A simple example would be the ring of ($ n \times n $)-matrices over $ 2 \mathbb{Z} $. This is a non-commutative ring without an identity.