defs 4

Question 1

Factor ring R/I

Answer 1

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)

Question 2

canonical surjection

Answer 2

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

Question 3

Fundamental Isomorphism Theorem

Answer 3

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

R/ker(θ) isomorphic to im( θ)

Question 4

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

Answer 4

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

Question 5

ideal of ring being maximal

Answer 5

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

Question 6

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

Answer 6

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