Give an example of a commutative ring without zero-divisors that is not an integral domain.

Q: Show that a unit of a ring divides every element of the ring.

A: Let us assume a unit in a ring R as a. Let us assume any element as x in that ring R.  By using the…

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: (B) Explain the relationship between am c) Boolean ring and commutative ring. d) Field and integral…

A: We have to find the relationship between boolean ring and commutative ring  Field and integral…

Q: Show that subring of an Artinian ring may not be Artinian.

A: According to the given information, it is required to show that subring of an Artinian ring may not…

Q: The number of zero divisors of the ring Z4+Z4 is?

A:

Q: show that the ring of complex umbers (+) Can be imbedding a ring of 2x2 matrics (M ₂x₂ 2X2

A: Given: A ring of complex numbers ℂ, +, · and a ring of 2×2 matrices M2×2, +, ·. To show: ℂ, +, · is…

Q: Prove that a division ring has no zero divisors.

A: Answer: Proof:

Q: Define principal ideal ring.

A: Ideal: A subring A of a ring R is called a (two-sided) ideal of A if for every r∈R and every a∈A…

Q: Determine all ring homomorphism from Z6 to Z6

A: Z6 is a cyclic, f is completely determined by f(1).

Q: Define reducible and irreducible elements in ring R.

A: Remark: An integral domain is a commutative ring with unity which does not contain any…

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: Show that a homomorphism from a field onto a ring with more thanone element must be an isomorphism.

A: Solution:

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

A:

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: Show that no finite field is algebraically closed.

A:

Q: The ring (Za. +3i3) can be imbedding in the ring (Z12-+12'12). True False

A:

Q: (B) Prove that: 1. Every Boolean ring is commutative. 2. Every field is integral domain.

A:

Q: Let R be a commutative ring with identity. Is it possible for R[x] to be a PID without being a…

A: Yes , it is possible for R[x] to be a PID ( assuming R[x] is PID ) without being a Euclidean domain.

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: Show that each homomorphism from a field to a ring is either one to one or maps everything onto 0.

A: Let ϕ:F→R  be a ring homomorphism from the field F to ring R . Now, the  kernel of ϕ is ideal of F.…

Q: Show that a commutative ring with the cancellation property (under multiplication) has no…

A:

Q: Prove: A field is an integral domain

A:

Q: Give an example of a finite noncommutative ring. Give an exampleof an infinite noncommutative ring…

A: A ring R is a set with two binary operations, addition and multiplication s.t. for all a,b,c in R:…

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: (B) Give an example for a commutative ring with identity.

A:

Q: Abstract Algebra

A: To define the concept of a subfield of a field and prove the stated property regarding subfields of…

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: Let (R,+,.) be acommutive ring with identity and a∈R be an idempotent which is different from 0 or 1…

A:

Q: can be embedded in a ring of endomorphisms of some Prove that a ring with unity abelian group.

A:

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

A:

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: Give an example of a noncommutative ring with unity.

A:

Q: Let R be the set of all functions. It is known that (R,+,×) is a ring where + and × are both…

A:

Q: If R is a commutative ring with unity, show that the characteristic of R[x] is the same as the…

A: If R is a commutative ring with unity, show that the characteristic of R[x] is the same as the…

Q: Make three different examples of a ring homomorphism which is one- one but not onto.

A: Make three different examples of a ring homomorphism which is one- one but not onto.

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

A: We know that

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

A:

Q: Show that a finite extension of a finite field is a simple extension

A:

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: The number of zero divisors of the ring Z4 + Z5 is

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

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

A: We will prove that the set of all elements in a ring R that are not zero divisors is closed under…

Q: Suppose that R is a commutative ring without zero divisors . show that characterstic of R is 0 or…

A: suppose that R is a commutative ring without zero divisors to show- characteristic of Ring R is 0 or…

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: (B) Define the integral domain ring. Is the product of integral domain rings also an integr domain?

A: We will define integral Domain ring.

Q: Prove that a field has no zero divisors.

A: