Question: Let R be a commutative ring and let M be an R-module. For a subset S M. The annihilator of S is defined to be

Let R be a commutative ring and let M be an R-module. For a subset S M. The annihilator of S is defined to be Ann(S) := {r R | x S, rx = 0}. Prove that Ann(S) is an ideal of R.

Step by Step Solution

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock blur-text-image

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Students Have Also Explored These Related Mathematics Questions!

Let R be a commutative ring and let M be an R-module. Let N be a submodule of M. Prove that the scalar multiplication of R on the quotient module M/N is well-defined.

6. (i) Show that $N=\{12,14,21\}$ generates $\mathbb{Z}$ as a $\mathbb{Z}$-module. [2 marks] (ii) Decide whether or not a proper subset of $N$ from part (i) [3 marks] generates $\mathbb{Z}$ as a...

6. (i) Show that $N=\{12, 14, 21\}$ generates $\mathbb{Z} $ as a $\mathbb{Z} $-module. [2 Marks] (ii) Decide whether or not a proper subset of $N$ from part (i) [3 Marks] generates $\mathbb{Z}$ as a...

6. (i) Show that $N=\{12,14,21\}$ generates $\mathbb{Z}$ as a $\mathbb{Z}$-module. [2 marks] (ii) Decide whether or not a proper subset of $N$ from part (i) [3 marks] generates $\mathbb{Z}$ as a...

6. (i) Show that $N=\{12,14,21\}$ generates $\mathbb{Z}$ as a $\mathbb{Z}$-module. [2 marks] (ii) Decide whether or not a proper subset of $N$ from part (i) [3 marks] generates $\mathbb{Z} $ as a...

Questions of Abstract Algebra: th & W N = =t . Let H be a subgroup of G of index 2. Prove that H is a normal subgroup of G. . Find the order of the group of rigid motions of a cube. . Give the...

Let R be a commutative ring and let S be any subset of R. Show that annihilator of S, Ann(S)={r is element of R | rs=0, for all s is element of S}, in an ideal of R.

Modern algebra: well written and well explanatory pls Let R be a commutative ring wilk identity, I an ideal of R and AIR- R/ I the canonical projection . a If S is a multiplicative subset of R, then...

Let R be a commutative ring with identity, I an ideal of R and : R R/ I the canonical projection. (a) If S is a multiplicative subset of R, then S = (S) is a multiplicative subset of R/ I. (b) The...

Every torsion-free divisible abelian group D is a direct sum of copies of the rationals Q. if O n e Z and D, then there exists a unique b D such that nb = a. Denote h by (1/n)a. For m, n Z (n ...

Richard and Karen, both 32, last month bought their dream house in Hamilton, Ontario. The purchase price was $550,000 plus additional fees such as land transfer tax, legal fees, administration fees...

Do some research and find what you would consider your fixed monthly payments when you graduate. The main ones I want you to consider is college loan, car payment and cell phone bill. If you pay...

What is time - series regression? Mitiple Cholce A mehod of cost estimation for a particular cost object based on information on other cost objects and variables, wif Used to describe regression...

I need help with task 1, can anyone help me solve it? Task 1: Use PCA from previous assignment to convert data from 4D to 2D. If you did not manage to implement PCA, you can simply use first 2...