Some Characterizations of <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1"> <mi>w</mi> </math>-Noetherian Rings and SM 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...

Some External Characterizations of SV-Rings and Hereditary Rings

In ring theory, the notion of annihilator is an important tool for studying the structures. Many characterizations and structure theorems can be derived by using this notion. On the other hand, certain classes of rings (e.g., Baer rings and Rickart rings) are defined by considering annihilators ideals. In the present work, we introduce a class of rings which is close to the class of Rickart rin...

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...