6. If a and b are not zero divisors in a ring R, prove that ab is not a zero divisor.

Q: 7. Prove that the set of all elements in a ring R that are not zero divisors is closed under…

A: Bb

Q: 7. Suppose that (R,+,.) be a commutative ring with identity and (I,+,.) be an ideal of R. If I is…

A: Given that (R, +, .) be a commutative ring with identity and (I, +, .)  be an ideal of R such that I…

Q: 6. Suppose that (M,+,.) be a maximal ideal of the commutative ring with identity (R, +,.) and x E M,…

A: Since, given that (M, + .) Is a maximal ideal of the commutative ring with identity (R, +, .)  and…

Q: Let (?,+, ⋅) be a ring with additive identity 0. Prove that for all x∈?, 0⋅x=0 and x⋅ 0 = 0.

A: We know that if (R,+,.) is a ring then (R,+) is an abelian group. And in abelian group, cancellation…

Q: * (10 Let (R, +,) be commutative ring with identity. Then a - b = a-c-b=c ifand only if R has no…

A: True

Q: Verify in the following statement involving the ideal generated by (a1, a2 , ... , ak ) in the ring…

A: To show that 2,3=ℤ Let 2,3=2x+3y: x,y∈ℤ -------(1) gcd2,3=1 , Thus 2m+3n=1 for some…

Q: Q:Let S and Szare two subrings of a ring (R, +..), prove that S, US2 subring of R iff either S, C S…

A:

Q: The number of zero divisors of the ring Z, Z5 is О 1 O 5

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: Verify in the following statement involving the ideal generated by (a1, a2, ... , ak )) in the ring…

A: To prove that: (2, 5)=ℤ

Q: 7. Suppose that (R, +..) be a commutative ring with identity and (I, +,.) be an ideal of R. If I is…

A:

Q: Let R be a ring with unity. Show that {а)3D( 2 хау: х, у є R}. finite

A:

Q: 16. If R is a field, show that the only two ideals of R are {0} and R itself.

A:

Q: 9. Suppose that (R,+, .) be a commutative ring with identity and x E rad R, then ..... .... (a) (x)…

A:

Q: Let R is ring of real numbers and *, (2) O defind on R^2 as follows, (a, b), (c, d)ER^2 then (a, b)*…

A:

Q: 9. Suppose that (R,+,.) be a commutative ring with identity and x E rad R, then (a) (x) = R (b) 1 —…

A:

Q: Let R is ring of real numbers and *, (2) O defind on R^2 as follows, V (a, b), (c, d)ER^2 then (a,…

A:

Q: 9. Suppose that (R,+, .) be a commutative ring with identity and xE rad R, then (a) (x) = R (b) 1-x…

A:

Q: 5. An element x in a ring R is called idempotent if a2 = x. Prove that if a is an idempotent element…

A:

Q: 4. Prove that a zero divisor in a ring cannot be a unit.

A:

Q: 5) Suppose that (R, +,.) be a ring without identity and has a subring with identity, then (a) R…

A:

Q: Show that in a Boolean ring R, every prime ideal P is not equal to R is maximal.

A:

Q: prove or disprove that the smallest non commutative ring is of order 4

A: Using the Result : " All rings of order p2 ( p is any prime) are commutative"   As 4=22 and 2 is a…

Q: In a ring R without unity, show that every idempotent is a zero divisor but is not nilpotent.

A:

Q: prove that the rings (R,+,.) and (Q,+,.) are fields.

A:

Q: Suppose that a and b belong to a commutative ring and ab is a zero-divisor.Show that either a or b…

A:

Q: Suppose that R is a commutative ring with unity 1. Then if ab is a zero divisor then a or b is a…

A:

Q: Prove or disprove that Q(√3) and Q(√-3) are ring-isomorphic.

A:

Q: The number of zero divisors of the ring Z4 Z2 is O 5 O 1

A:

Q: lisprove: that the triple <Z,,, constitutes a ring. et of units U(Z₂), and state whether (U(Z₁), Ⓒ)…

A:

Q: The number of zero divisors of the ring Z4 O Z, is

A:

Q: Show that if n is an integer and a is an element from a ring, thenn . (-a) = -(n . a).

A:

Q: B. Show that each homomorphism from a field to a ring is either one to one or maps everything nnto 0…

A: 18 Suppose we have a homomorphism φ : F → R where F is a field and R is a ring (for example R itself…

Q: Show that in a ring R cancellation laws hold if and only if R has no zero divisors.

A:

Q: Prove or dlspive that the subset M: ta+3biabeZ5 IS a subring of the Causslan integer ring ZCiJ

A: If R is a ring  and S is a subset of R is said to be a  subring if it is closed in the following…

Q: Let R be a commutative ring, a, b e R and ab is a zero-divisor. Show that either a or b is a…

A: We have to solve given problem:

Q: Suppose that R is a ring with no zero-divisors and that R contains anonzero element b such that b2 =…

A: Given Suppose that R is a ring with nonzero-divisors and that R contains a nonzero element b such…

Q: et R be a ring and assume a∈R is not a zero divisor. If ab=ac for some b,c∈R, then b=c

A:

Q: 9. Suppose that (R,+,.) be a commutative ring with identity and x E rad R, then (a) (x) = R (b) 1- x…

A:

Q: Suppose that R is a commutative ring without zero-divisors. Showthat all the nonzero elements of R…

A: Given: R is a commutative ring without zero-divisors. To show: all the nonzero elements of R have…

Q: 5. Let A and B be two ideals of a commutative ring R ith unity such that A + B=R. Show that AB =…

A:

Q: 9. Suppose that (R,+,.) be a commutative ring with identity and x E rad R, then (a) (x) = R (b) 1 —…

A:

Q: Let a and b be elements in a ring R. If ab is a zero divisor, prove that either a or b is a zero…

A: Given :          ab is a zero divisor where a and b are elements in ring R. Formula :        Zero…

Q: Show that the factor ring Q[r] (P(x)) has a zero-divisor

A:

Q: Suppose I,J be ideals of a commutative ring R. Prove that IJ cInJ.

A:

Q: 2. Let R be a ring, and a, b e R. Prove exactly one of the following statements: (a) If ab is a left…

A: As exactly one statement is required to be proved. I'll prove (b) Statement.

Q: 14) Any subring of a commutative ring is commutative. True False

A: I have given an answer in step 2.

Q: 4) In a ring R; The sum of two non-trivial idempotent elements is not always an if we take + = is…

A: 4) We need to show that , sum of two non trivial idempotent elements is not always idempotent. We…

Q: 5. Give an example where a and b are not zero divisors in a ring R, but the sum a +b is a zero…

A:

Q: Verify in the following statement involving the ideal generated by (a1, a2, ... , ak ) in the ring…

A: we have to prove,(10,15)=(5)

Q: 5. Let A and B be two ideals of a commutative ring R with unity such that A +B = R. Show, that AB=A…

A: