Definition of Ring Vs Rng

Question

When I took abstract algebra I learned that a ring was a set that is an abelian group under addition and monoid under multiplication (along with the distributive property).

In preperation to tutor someone in algebra I've noticed that some books present a ring as what I know as a "rng" or an abelian group under addition and a semi-group under multiplication.

Is there any reason to prefer one as the definition for a ring vs the other?

EDIT And a very related question, is there any math authority or consensus that has dictated/specified that it is more correct to use the ring/ring with unity or the rng/ring definition?

Answer 1

There are no math authorities. There are just conventions, and as far as I can tell the convention that "ring" means "ring without identity" can only be traced back to people who learned algebra using Hungerford.

The main reason to prefer "ring" to mean "ring with identity" is that I am pretty sure it is the statistically dominant convention, although I don't have the statistics to actually back that up. (Unless this is not what you mean by "reason," in which case I'll guess another possible meaning: for most applications, your rings will have identities.)

Answer 2

The standard definition of a ring $R$ is that $R$ is an abelian group under addition and a semi-group under multiplication. The existence of multiplicative identity is generally not required. If rings are defined in the "monoid under multiplication" way we can not consider the set $\{0\}$ as the smallest possible ring. (I see Artin defines rings in the "monoid under multiplication" way where Dummit & Foote, Herstein do not. So there is inconsistency in the standard definition.)

Answer 3

Wikipedia had a large discussion of this from 2003 to 2008 including an analysis of publications, and comments about both Bourbaki and Cambridge University changing to require a 1.

There does not seem to be a consensus, but there does seem to be a trend towards more modern and more advanced texts being more likely to require a 1.