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Commutative Algebra Homework 2
School: UCLA
Math538 Commutative Algebra Homework 2 Name: Zhaoning Yang September 20, 2013 Problem 1 Suppose that W1 W2 are multiplicative sets in a ring R and M is an Rmodule. Show 1 1 1 that W2 (W1 M ) W2 M . Solution: 1 1 1 W2 R by the universal property. So, we c

Study Notes CommAlgebra
School: UCLA
Math 538 Commutative Algebra Study Notes Zhaoning Yang January 4, 2014 1 Rudiments 2 Nakaymas Lemma Denition 2.1. Let R be a ring, be an index set, and M be an Rmodule for each . we dene (1) (2) M = cfw_(m )  m M to be the direct product of M M M t

Homework 1 Solutions
School: UCLA
Math 538 Commutative Algebra Homework 1 Name: Zhaoning Yang September 4, 2013 Problem 1 Let R be a ring, if I, J, J are ideals in R, prove that I J J implies that either I J or I J . If P is a prime ideal in R, then show that I J, I J , or I P . Solution:

Homework 4
School: UCLA
Math 538 Commutative Algebra Homework 4 Name: Zhaoning Yang October 18, 2013 Problem 1 Let R be an integral domain and K(R) be its eld of fractions. We dene the normalization of R to be the integral closure of R in K(R), which is denoted by RN . We say th

Homework 3
School: UCLA
Math 538 Commutative Algebra Homework 3 Name: Zhaoning Yang October 4th, 2013 Problem 1 Suppose R is a ring and M, M are Rmodules. Prove that M M is at if and only if M and M are individually at. Solution: Proof. Let M, N and P be Rmodules, from construc

Homework9
School: UCLA
Math 210A: Algebra, Homework 9 Ian Coley December 3, 2013 Problem 1. Determine all subrings of Z. Solution. Let R Z be a subring. Then we must have 1 R. Since R is closed under addition, we have 1 + . . . + 1 = n R. Since R is closed under taking additive

Homework8
School: UCLA
Math 210A: Algebra, Homework 8 Ian Coley December 1, 2013 Problem 1. Show that if 1 = 0 in a ring R, then R is the zero ring. Solution. Let x R be an element. Then x = 1 x = 0 x = 0. Therefore R is the zero ring. Problem 2. (a) For any ring R, dene a ring

U2
School: UCLA
Unit II: Matrix Algebra 1. Algebraic Operations with Matrices addition of matrices, multiplication of a matrix by a scalar Definition of product of matrices: If A is an m n matrix, and B is an n p matrix with columns b1 , . . . , bp Rn , then AB i