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

Q: Let R be a euclidean ring and a,b are non zero elements of R.if a and b are associates,show that…

A: Let R be a Euclidean domain with measure d. Also given that a and b are associates in R. Then from…

Q: The image of a commutative ring under homomorphism may be noncommutative False True

A: As you asked multiple questions , I solved the first one. So you asked "the image of a commutative…

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

A: Bb

Q: The number of zero divisors of the ring Z, Zg is 11 O O O

A: Consider the provided question, We need to find number of zero divisors of the ring Z4⊕Z5. Z4⊕Z5≈Z20…

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

A:

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: Let a and b be idempotents in a commutative ring. Show that eachof the following is also an…

A:

Q: A ring (R. +.) .) is commutative if addition is commutative in R. O True O False

A: Solve the following

Q: An example on a finite non- commutative ring without identity is:

A: Matrices are noncommutative  So we can choose set of matrices  Not if we take Mn[R] then this is non…

Q: Decide whether ZxZ = {(n,m) n,me Z} with addition and multiplication by components, give a ring…

A:

Q: Give an example of a noncommutative ring that has exactly 16 elements.  Prove that it is a…

A: Ans:   Let R=M2 Z2.This is a finite 16 elements   non commutative ring with identity…

Q: If a is an idempotent in a commutative ring, show that 1 - a is alsoan idempotent.

A: Here given

Q: The number of zero divisors of the ring Z4 O Z3 is

A: We have to find the number of zero divisors in the ring Z4⊕Z3.

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

A:

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

A:

Q: Every element in a ring has an Multiplicative inverse?

A: A set R is said to be ring with two binary operations addition and multiplication, if R is…

Q: Give an example to show that the factor ring of a ring with zero divisors may be an integral domain.

A: (i)4ℤ8ℤ≈ℤ2 (ii) ℤ6ℤ

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

A:

Q: Find elements a, b, and c in the ring Z ⨁ Z ⨁ Z such that ab, ac, andbc are zero-divisors but abc is…

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: Let CR,t,,) be a Commutative ring ?

A: First I recall you Definition of commutative ring.  A commutative ring is ring in which…

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

A:

Q: Construct the adition and multiplication tables for the ring Z/4Z.

A: Given: A ring ℤ/4ℤ. To construct: Addition and multiplication tables for the ring ℤ/4ℤ.

Q: Determine all reversible elements in ring Z_6 and Z_8 and give their inverse

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: Construct five different examples of left ideals of a ring which are not right ideals.

A:

Q: If A and B are ideals of a commutative ring R with unity and A + B= R, show that A N B = AB.

A: Given Data: A and B are ideal of a commutative ring R with unity And A + B = R   For A⋂B = AB, The…

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

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: Show if R is a commutative ring then R[x] is also a commutative ring.

A:

Q: Give an example of a noncommutative ring with unity.

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: The number of zero divisors in the ring Z100 is:

A: We know that

Q: Write the multiplication table for the following commutative quotient ring: Z2[x]/(x³ + x² +x + 1).

A: Given that, ℤ2x/x3+x2+x+1 is a commutative quotient ring. We have to frame the multiplication table…

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: Determine whether (Z,⊕,∗) is a ring with binary operations defined as (x⊕y)=x+y−7,…

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: Is the multiplicative inverse of an element in a polynomial ring unique?  Or can there be more than…

A: This can be proved as shown below:

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

A:

Q: Which one of the following is not a commutative ring? (a) (Q. +, :) (b) (Z5, O5, 85) (c) (Z12, O12,…

A:

Q: Prove that a finite ring R with unity and no zero divisors is a division ring.

A:

Q: Show, by example, that for fixed nonzero elements a and b in a ring, the equation ax = b can have…

A: Solution:-           consider the ring Z4         Let a=2 &  b=2:         Then 21=2 &…

Q: Determine all elements of a ring that are both units and idempotents

A: To determine: All elements of a ring that are both units and idempotent.

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

A: I have given an answer in step 2.

Q: Let a, b, and c be elements of a commutative ring, and suppose thata is a unit. Prove that b divides…

A: Let a, b, and c be elements of a commutative ring where a is a unit. Suppose that bdivides c. Then c…

Q: The number of idempotents elements in the ring Z6 is: 1 8 4

A: Ans : 4  Last option is true List of idempotent element is {0,1,3,4}

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: show  that a 2 by2 matrix is a  ring and further is non cummutative

A: