SQT

>>9104141
>would the cardinality be 125?
Why do you think it is?

>>9104141

If I'm interpreting the problem correctly, then R has 5^4 = 625 elements, while I has 5^1 = 5 elements. Cursory review also shows that the leq symbol as used in a) indicates that I is a subgroup of R (that is, a proposition whose truth value is to be determined), the triangle-eq symbol of b) means that (presumably) I is a /normal/ subgroup of R, and the R/I notation denotes the so-called factor group set, or quotient group set, given by R/I = {aI | a in R}, the very notation's definition being contingent on I actually /being/ a normal subgroup of R, in my book at least - and presumably motivating b), which context would strongly suggest is to be answered in the affirmative.

So, do we have that I <= R? Now I wonder a bit because ring entails two operations, while a group entails just one. It's also possible that I've misinterpreted the above notations a bit.

>>9104163
Close, but the notation is probably a) checking if I is a subring of R and the notation in b) checking if I is an ideal of R and c) analysing the quotient ring

>>9104167

And upon review of the definition of an ideal, the various questions basically port 1-1 from the group notions which I'd just suggested, to their ring context.

Different authors use different conventions. I own four books about abstract algebra. /Every single one/ has a materially distinct definition of a field.

Hmmm, what would the coset representatives be?