Question

Let R be a commutative integral domain as defined in [H3, p. 117]. An element x of a module X over R is said to be a torsion element of X iff there is a nonzero element ? ? R with ?x = 0. Prove that the torsion elements of X form a submodule ?(X) of X. ?(X) is known as the torsion submodule of X. A module X is said to be a torsion module iff ?(X) = X and to be torsion-free iff ?(X) = 0. For any module X over R, prove that ?(X) is a torsion module. Also prove the following statements: (a) R is a torsion-free module over itself. (b) Every submodule of a torsion module over R is itself a torsion module over R. (c) Every submodule of a torsion-free module over R is itself a torsion-free module over R.

          Let R be a commutative integral domain as defined in [H3, p. 117]. An element x of a module X over R is said to be a torsion element of X iff there is a nonzero element ? ? R with ?x = 0. Prove that the torsion elements of X form a submodule ?(X) of X. ?(X) is known as the torsion submodule of X. A module X is said to be a torsion module iff ?(X) = X and to be torsion-free iff ?(X) = 0. For any module X over R, prove that ?(X) is a torsion module. Also prove the following statements:
(a) R is a torsion-free module over itself.
(b) Every submodule of a torsion module over R is itself a torsion module over R.
(c) Every submodule of a torsion-free module over R is itself a torsion-free module over R.

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Let R be a commutative integral domain as defined in [H3, p. 117]. An element x of a module X over R is said to be a torsion element of X iff there is a nonzero element ? ? R with ?x = 0. Prove that the torsion elements of X form a submodule ?(X) of X. ?(X) is known as the torsion submodule of X. A module X is said to be a torsion module iff ?(X) = X and to be torsion-free iff ?(X) = 0. For any module X over R, prove that ?(X) is a torsion module. Also prove the following statements:
(a) R is a torsion-free module over itself.
(b) Every submodule of a torsion module over R is itself a torsion module over R.
(c) Every submodule of a torsion-free module over R is itself a torsion-free module over R.

Added by Rosa H.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Madhur L

11/19/2023

Step 1

We need to show that the torsion elements of X form a submodule T(X) of X. To do this, we need to show that T(X) is closed under addition and scalar multiplication. Let x, y be torsion elements in X. Then there exist nonzero elements a, b in R such that ax = 0 Show more…

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Let R be a commutative integral domain as defined in [H3, p. 117]. An element x of a module X over R is said to be a torsion element of X iff there is a nonzero element λ ∈ R with λx = 0. Prove that the torsion elements of X form a submodule τ(X) of X. τ(X) is known as the torsion submodule of X. A module X is said to be a torsion module iff τ(X) = X and to be torsion-free iff τ(X) = 0. For any module X over R, prove that τ(X) is a torsion module. Also prove the following statements: (a) R is a torsion-free module over itself. (b) Every submodule of a torsion module over R is itself a torsion module over R. (c) Every submodule of a torsion-free module over R is itself a torsion-free module over R.

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Transcript

00:02 Hello, we are given that r is a commutative integral domain, x is a module over r and if x is element of x such that lambda times x equal to 0 for some non -zero element lambda of r then x is called a torsion element of x and the set of all torsion elements of a module x is denoted by the symbol tau of x.

00:27 First we have to prove that the set tau of x is a sub module of x.

00:30 So we have to prove that tau of x is closed under scalar multiplication and addition.

00:36 So let x1, x2 be arbitrary torsion elements of x that is arbitrary elements of tau x.

00:42 Then lambda 1 x1 equal to 0 and lambda 2 x2 equal to 0 for some non -zero elements lambda 1 lambda 2 equal to lambda for some non -zero element lambda 2 elements lambda 1 lambda 2 of r.

00:52 But then lambda 1 times lambda 2 is also not equal to 0 since each of them is not 0 and r is an integral domain.

00:59 This implies that lambda 1, ok we have this.

01:04 Now note that lambda 1 lambda 2 times x1 plus x2 is lambda 1 lambda 2 times x1 plus lambda 1 lambda 2 times x2 using commutativity of r and associativity of multiplication, scalar multiplication we have that lambda 1 lambda 2 x1 equal to lambda 2 lambda 1 x1.

01:23 Now lambda 1 x1 is equal to 0 so this product is 0.

01:28 Similarly this product we can write like this and lambda 2 x2 is equal to 0 so this product is equal to 0.

01:33 So the right hand side is equal to 0 is sum of 0 plus 0 so it's 0.

01:38 So we have that lambda 1 lambda 2 which is a non -zero element of r times x1 plus x2 is 0...

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