I'm trying to make sense of two different definitions of an algebra over a ring. The definitions are as follows:(adsbygoogle = window.adsbygoogle || []).push({});

If R is a commutative ring, then

1) S is an R-algebra if S is an R-module and has a compatible ring structure (such that addition agrees)

2) If [itex] \alpha:R \to S [/itex] is a ring homomorphism such that [itex] \alpha(R) [/itex] is in the centre of S, then [itex] \alpha [/itex] is an R-algebra. We normally abuse notation in this instance and say that S is the R-algebra when [itex] \alpha [/itex] is understood.

I think the second definition is more technically valid, albeit more complicated. So my first question: Via the first definition, are all commutative rings R also R-algebras? It seems like this would be true, since R is an R-module over itself and it certainly has a compatible ring structure.

Secondly, I want to make sure we can get from the second definition the first. I think I can do it as follows:

First, the ring homomorphism [itex] \alpha:R \to S [/itex] defines an R-module structure on S via the map [itex] \rho: R\times S \to S, \rho(r,s) = \alpha(r)s [/itex]. Next, since [itex] \alpha(r)s = s \alpha(r) [/itex] for all r and s and [itex] \alpha [/itex] is a homomorphism, then

[tex] [\alpha(r_1)s_1 ][\alpha(r_2)s_2] = \alpha(r_1r_2)s_1 s_2. [/tex]

This last step seems right, but in my head I can't figure out why this gives compatibility with the ring structure of S.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Rings as algebras

**Physics Forums | Science Articles, Homework Help, Discussion**