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 -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 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.
A nonunital ring or rng is a set with operations of addition and multiplication, such that:
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 .
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 .
A survey of commutative rng theory is in
Discussion of module theory over rngs is in
The notation “rng” originates in
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”.