Monthly Archives: October 2012

Introduction to Ring Theory (8)

Matrix Rings In this post, we’ll be entering the matrix. Let R be a ring. The ring Mn×n(R) is the set of matrices whose entries are elements of R, where the addition and multiplication operations are given by the usual matrix addition … Continue reading

Posted in Notes | Tagged , , , , , , | Leave a comment

Introduction to Ring Theory (7)

Polynomial Rings A polynomial over a ring R is an expression of the form: , where , and . Let’s get some standard terminology out of the way. The element ai is called the coefficient of xi. The largest n for which an ≠ 0 is called … Continue reading

Posted in Notes | Tagged , , , , , , | Leave a comment

Introduction to Ring Theory (6)

Let’s keep stock of what we’ve covered so far for ring theory, and compare it to the case of groups. There are loads of parallels between the two cases. G is a group R is a ring. Abelian groups. Commutative … Continue reading

Posted in Notes | Tagged , , , , | Leave a comment

Introduction to Ring Theory (5)

Our first order of the day is to state the correspondence between the ideals and subrings of R/I and those of R. This is totally analogous to the case of groups. Theorem. Let I be an ideal of R. There are 1-1 … Continue reading

Posted in Notes | Tagged , , , , , , | Leave a comment

Introduction to Ring Theory (4)

It’s now time to talk about homomorphisms. Definition. Let R, S be rings. A function f : R → S is a ring homomorphism if it satisfies the following: f(1R) = 1S; f(x+y) = f(x) + f(y) for all x, y in … Continue reading

Posted in Notes | Tagged , , | Leave a comment

Introduction to Ring Theory (3)

Ideals and Ring Quotients Suppose I is a subgroup of (R, +). Since + is abelian, I is automatically a normal subgroup and we get the group quotient (R/I, +). One asks when we can define the product operation on R/I. To be specific, each … Continue reading

Posted in Notes | Tagged , , , , , , | Leave a comment

Introduction to Ring Theory (2)

Subrings Just like groups have subgroups, we have: Definition. A subset S of a ring R is a subring if it satisfies the following: ; ; . The first two conditions imply that S is a subgroup of (R, +). Together with … Continue reading

Posted in Notes | Tagged , , , | Leave a comment