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 R-module for each . we dene
(1)
(2)
M
= cfw_(m ) | m M to be the direct product of M
M
M t
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
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 R-modules, from construc
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 R-module. Show
1
1
1
that W2 (W1 M ) W2 M .
Solution:
1
1
1
W2 R by the universal property. So, we c
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: