Factor ring R/I

Let R be a ring and let I be a proper ideal. Let R/I denote the set of cosets of I in the additive group < R, + >

R/I = {r + I | r ∈ R}.

Define operations + and × on R/I by:

(r + I) + (s + I) = (r + s) + I

(r + I) × (s + I) = (r × s) + I

(so we’re adding and multiplying cosets, these being the elements of R/I)

canonical surjection

The map π : R → R/I defined by π(r) = r + I is a surjective ring homomorphism with kernel I

Fundamental Isomorphism Theorem

Let θ : R → S be a ring homomorphism. Then

R/ker(θ) isomorphic to im( θ)

Let I be an ideal of the ring R. Then the set of ideals of R which contain I is in one-to-one correspondence with the set of ideals of the factor ring R/I:

- to an ideal J ≥ I there corresponds ……
- to an ideal K

Let I be an ideal of the ring R. Then the set of ideals of R which contain I is in one-to-one correspondence with the set of ideals of the factor ring R/I:

- to an ideal J ≥ I there corresponds πJ = {r + I | r ∈ J} = {π(r) | r ∈ J}, an ideal in R/I;
- to an ideal K

ideal of ring being maximal

An ideal I of a ring R is maximal if it is proper (i.e. I =! R)

and for any ideal J with I ≤ J ≤ R, then J = I or J = R

A proper ideal I of a commutative ring R is prime ..

A proper ideal I of a commutative ring R is prime if whenever r, s ∈ R and rs ∈ I then either r ∈ I or s ∈ I