Question

In: Advanced Math

Suppose a and b belong to a commutative ring and ab is a zero-divisor. Show that...

Suppose a and b belong to a commutative ring and ab is a zero-divisor. Show that either a or b is a zero-divisor. provide explanation please.

Solutions

Expert Solution

Recall the following defintions:

Let be a ring.

(i) is called commutative if .

(ii) An element is called a zero divisor if there is a non-zero element such that either or .

Back to the problem:

Suppose is a commutative ring and such that is a zero divisor.

Then there exists a non-zero element such that either or . Here both the conditions are same because of the commutativity.

If is non-zero then from the definition of a zero divisor it follows that is a zero divisor.

If then from the definition of a zero divisor it follows that is a zero divisor because here is non-zero. (Proved)

Colby Messinger answered 1 year ago

Next > < Previous

Related Solutions

Given a commutative ring with identity R. A. Prove that a unit is not a zero...

Given a commutative ring with identity R. A. Prove that a unit is not a zero divisor. B. Prove that a zero divisor cannot be a unit.

Let A be a commutative ring and F a field. Show that A is an algebra...

Let A be a commutative ring and F a field. Show that A is an algebra over F if and only if A contains (an isomorphic copy of) F as a subring.

show that for some ring R, the equality a2−b2=(a−b)(a+b) holds ∀a,b∈R if and only if R is commutative.

show that for some ring R, the equality a^2−b^2=(a−b)(a+b) holds ∀a,b∈R if and only if R is commutative.

Problem 1. Suppose that R is a commutative ring with addition “+” and multiplication “·”, and...

Problem 1. Suppose that R is a commutative ring with addition “+” and multiplication “·”, and that I a subset of R is an ideal in R. In other words, suppose that I is a subring of R such that (x is in I and y is in R) implies x · y is in I. Define the relation “~” on R by y ~ x if and only if y − x is in I, and assume for the...

Problem 2. Suppose that R is a commutative ring, and that R[X] := { (a_0, a_1,...

Problem 2. Suppose that R is a commutative ring, and that R[X] := { (a_0, a_1, a_2, ...)^T | a_i is in R, a_i not equal to 0 for only finitely many i} is the set of polynomials over R, where we have named one particular element X := (0, 1, 0, 0, . . .)T . Show that R[X] forms a commutative ring with a suitably-chosen addition and multiplication on R[X]. This will involve specifying a “zero” element of...

(3) Let m be a positive integer. (a) Prove that Z/mZ is a commutative ring. (b)...

(3) Let m be a positive integer. (a) Prove that Z/mZ is a commutative ring. (b) Prove that if m is composite, then Z/mZ is not a field. (4) Let m be an odd positive integer. Prove that every integer is congruent modulo m to exactly one element in the set of even integers {0, 2, 4, 6, , . . . , 2m− 2}

(a) Show that the length of the broken line satisfies Length(L) ≥ |AB|. (b) Show that...

(a) Show that the length of the broken line satisfies Length(L) ≥ |AB|. (b) Show that L achieves the lower bound Length(L) = |AB| if and only if the vertices V1,...,Vk−1 all lie on the segment AB and appear in that orderonAB,i.e.,theysatisfyVi ∈Vi−1Vi+1 forall1≤i≤k−1.