isomorphism

If R and S are rings then an isomorphism from R to S is a bijection θ : R → S such that, for all r, r’ ∈ R we have

• θ(r + r’) = θ(r) + θ(r’) and

• θ(r × r’) = θ(r) × θ(r’)

If θ is an isomorphism from R to S then we write θ : R ‘ S. We say that R and S are isomorphic

homomorphism

If R and S are rings then a homomorphism from R to S is a map θ : R → S such that, for all r, r’ ∈ R we have

• θ(r + r’) = θ(r) + θ(r’)

• θ(r × r’) = θ(r) × θ(r’)

• θ(1R) = 1S

embedding/monomorphism

An embedding, or monomorphism, is an injective (i.e. one-to-one) homomorphism

kernel

If θ : R → S is a homomorphism of rings then the kernel of θ, ker(θ), is the set, {r ∈ R | θ(r) = 0}, of elements which θ sends to 0S

automorhism

An automorphism of a ring is an isomorphism from the ring to itself

ideal

An ideal of a ring R is a subset I ⊆ R such that:

• 0 ∈ I;

• a + b ∈ I, for all a, b ∈ I;

• ar ∈ I and ra ∈ I, for all a ∈ I and for all r ∈ R.

write I

principal ideal generated by a

If a ∈ R then {r1as1 + · · · + rnasn | n ≥ 1, ri , si ∈ R} is, you

should check, an ideal which contains a and is the smallest ideal of R containing a.

It is called the principal ideal generated by a and is denoted <a>. If R is commutative then its description simplifies: </a><a> = {ar | r ∈ R}.</a>

A principal ideal is one which can be generated by a single element.

</a>