Show that a commutative ring with the cancellation property (under multiplication) has no zero-divisors.

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: Give an example of a ring that has exactly two maximal ideals

A: To Do: Give an example of a ring that has exactly two maximal ideals.

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

A:

Q: Define commutative ring.

A: Answer:

Q: Show that a ring is commutative if it has the property that ab = caimplies b = c when a ≠ 0.

A:

Q: Construct the multiplication table for the ring Z/6Z.

A:

Q: Show that the nilpotent elements of a commutative ring form a subring

A: We have to prove that the nilpotent elements of a commutative ring form a subring. In order to do…

Q: Let R = Pn be the set of all polynomials in the unknown x of degree less than or equal to a fixed…

A: R=Pn be the set of all polynomial in the unknown x of degree less than or equal to to a fixed…

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

A:

Q: The number of zero divisors in the ring Z80

A: Step:-1 Theorem:- If  gcda, n>1  and n- a then [a] is a zero divisor in ℤn. Also, we know ϕn =…

Q: Enter the multiplicative inverse of the element [10] x + [13] in the ring z-[x]

A: We need to fin the multiplicative inverse of the element [10]20x+[13]20 in the ring Z20x. Let…

Q: Show that Corollary 2 of the Theorem is true over any commutativering with unity.

A: Let F be a field and a∈F. Here fx∈Fx. To show that a is a zero of fx if and only if x-a is the…

Q: If a and b are nilpotent element of a commutative ring, show that a+b is also nilpotent. Give an…

A:

Q: Explain why the ring of integers Z under usual addition and multiplication is not a field.

A: Given that the set of integers Z is a ring under addition and multiplication. We know that an…

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: Check if the following subsets of real numbers are Rings or not. • {a + b*(2)^(1/3) | a, b ∈ Q} • {a…

A:

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

A:

Q: Show that the system consisting of all elements of the form a+b√5, where a and b are any rational…

A:

Q: Show that every ring can be embedded in a ring with identity.

A: We have to show that every ring can be embedded in a ring with identity.

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

A: Integral domain: Let D be a ring. Then D is an integral domain, provided these conditions hold: 1. D…

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 homomorphic image of a commutative ring is a commutative ring.

A:

Q: I is set f all integers (positic, is the ordinaiy multiplication negative and zero),. and show that…

A:

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

A: I have given an answer in step 2.

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: 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: