symmetric monoidal (∞,1)-category of spectra
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 -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).
A nonunital ring or rng is a set with operations of addition and multiplication, such that:
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.
Given a non-unital commutative ring , then its unitisation is the commutative ring obtained by freely adjoining an identity element, hence the ring whose underlying abelian group is the direct sum of with the integers, and whose product operation is defined by
where for and we set .
One can also think of the unitisation as the quotient of the polynomial ring “” ( “with a generic element adjoined”, i.e. a term algebra) quotiented by the relations , so that this must be a right and left identity for multiplication . Cosets must be represented by expressions of the form ; this provides an obvious isomorphism to the above definition, and motivates the multiplication for above.
Since embeds into its unitisation , every rng lives as an ideal in some ring. We can consider -linear actions on abelian groups . 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 along to an action map . This induces an equivalence so that one may as well study if one wanted to study through its module category.
This is because the definition of any ring homomorphism out of is uniquely fixed on the -summand.
That the functor is fully faithful is to observe that for a ring the fiber over is a torsor over the additive group underlying the augmentation ideal , and moreover it is a pointed torsor, the point being itself, hence is canonically equivalent to the augmentation ideal , the equivalence being addition by in . Hence any homomrphism of rings with identity over
is uniquely fixed by its restriction to the augmentation ideal , whose image, moreover, has to be in the augmentation ideal .
The identification of non-unital algebras as augmentation ideals of augmented unital algebras is used for instance in (Fresse 06).
The terminology “rng” originates in
A survey of commutative rng theory 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