Let R be a commutative ring with identity. We will say that an ideal I of R is called "prime" is ab in I implies that either a is in I or b is in I.

Show that if the Ideal I is prime, then the quotient ring R/I is an integral domain.

Question

Let R be a commutative ring with identity. We will say that an ideal I of R is called "prime" is ab in I implies that  either a is in I or b is in I. 

Show that if the Ideal I is prime, then the quotient ring R/I is an integral domain.

amistre64 · Answer

ive seen that R/I notation before, but I havent seen a definition of it.

anonymous · Answer

its congruence class

amistre64 · Answer

i recall that term in modular arithmetic ...  can you explain how R/I defines a congruence class?

amistre64 · Answer

R means ring,  I means ideal ....  as opposed to my misreading R as "set of Reals"

anonymous · Answer

you're correct

anonymous · Answer

R is ring.

amistre64 · Answer

and an Ideal is an additive (left and right) subgroup of a ring?

amistre64 · Answer

hmm, does that mean that an ideal is a closed subgroup?

anonymous · Answer

A subring I of R is ideal if whenever r in R and a in I, then ra is in I and ar is in I

amistre64 · Answer

and an integral domain is just the domain of integers:  ...-3,-2,1,0,1,2,3,...

what is a "quotient ring"?

anonymous · Answer

the ring R/I is called the quotient ring of R by I (or factor ring). One sometimes speaks of factoring out the ideal I to obtain the quotient ring R/I

anonymous · Answer

quotient rings are the natural generalization of congruence-class arithmetic in Z and F[x].

amistre64 · Answer

id prolly have to spend a day or so reading up on this stuff to be useful on this one .... theres just too much new terminology for me to parse thru :/

anonymous · Answer

I'm working with the book "abstract algebra : an introduction" 2nd editon  Thomas W Hungerford. My final is tomorrow But I'm going down to my professors office in an hour. It's not a big deal if you can't help, but that's the book if you're interested in your own independent study.

amistre64 · Answer

ive got:  John B. Fraleigh, (2003), A First Course in Abstract Algebra, Seventh Edition that my teacher gave me to read over the summer :)  but ive been looking for other literature as well, thnx

anonymous · Answer

you know how when you get a book and it talks to you like you're a nuclear physicist and it gets annoying because it gives you things it already expects you to have a full understanding of? well this book does a very good job at introducing things and explaining them explicitly. I've just never been a very good mathematician.