Centralizing <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1"> <mi>n</mi> </math>-Homoderivations of Semiprime Rings

MATH 436 Notes: Rings

1 Rings Definition 1.1 (Rings). A ring R is a set with two binary operations + and · called addition and multiplication respectively such that: (1) (R, +) is an Abelian group with identity element 0. (2) (R, ·) is a monoid with identity element 1. (3) Multiplication distributes over addition: a · (b + c) = a · b + a · c and (b + c) · a = b · a + c · a for all a, b, c ∈ R. A ring is called commu...

MATH 436 Notes: Examples of Rings

i∈N R = {(a0, a1, a2, . . . )|aj ∈ R} is a countable direct product of (R,+) with itself. Thus the elements of R[[x]] are arbitrary sequences with entries in R where addition is componentwise. We define a multiplication ⋆ via (a0, a1, . . . ) ⋆ (b0, b1, . . . ) = (c0, c1, . . . ) where cn = ∑n k=0 akbn−k for all n ∈ N. It is easy to check that the sequence (1, 0, 0, . . . ) is a two-sided ident...

Math 8020 Chapter 1: Commutative Rings

MATH 436 Notes: Factorization in Commutative Rings

Proposition 1.1. Let f : R1 → R2 be a homomorphism of rings. If J is an ideal of R2, then f (J) is an ideal of R1 containing ker(f) and furthermore f(f(J)) ⊆ J . Now let f : R1 → R2 be an epimorphism of rings. If J is an ideal of R2 then f(f (J)) = J . If I is an ideal of R1 then f(I) is an ideal of R2. Furthermore we have I ⊆ f (f(I)) = I + ker(f) and thus I = f(f(I)) if I contains ker(f). Thu...

Math 254a: Rings of Integers and Dedekind Domains

Proof. Let (α1, . . . , αn) be any Q-basis for K; we first claim there for each i, there exists a non-zero di ∈ Z such that diαi ∈ OK . Indeed, it is easy to check that it is enough to let di be the leading term of any integer polynomial satisfied by αi. Thus, for any non-zero β ∈ I, we find that (βd1α1, . . . , βdnαn) is a Q-basis for K contained in I. It remains to show that such a basis with...