Let R be a commutative ring with more than one element. Provethat if for every nonzero element a of R we have aR = R, then R hasa unity and every nonzero element has an inverse.

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: Let R = {0, 2, 4, 6, 8}. Then R is a ring under addition and multiplication (mod 10). Find the…

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Q: Let R be a ring that contains at least two elements. Suppose for each nonzero a e R there is a…

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Q: If a is unit in a ring R then a is Select one: a. invertable b. the additive identity C. zerodivisor…

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

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Q: If R is a ring, then every element of R is either a unit or a zero-divisor. Select one: O True O…

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Q: A ring may have more than one identity element for multiplication? T or F

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Q: Let R be a commutative ring and a e R. Show that if a is nilpotent, then ab is nilpotent for each be…

A: Solution. Since a is nilpotent , an = 0 for some n ∈ N.

Q: Let R be a ring with unity and assume p, q, r ∈ R∗ Find (pqr)-1and also prove that it’s the inverse…

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Q: Let R and S be commutative rings. Prove that (a, b) is a zero-divisorin R ⨁ S if and only if a or b…

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Q: If R is a noncommutative ring with unity and x, y ∈ R, compute the product x(x + y)(x − y)y.

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Q: Is the ring Z/6Z and Z6 are isomorphic.

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Q: Let R = {0, 1, x}, where 0, 1, and x are distinct, and let (R, +, ⋅) be a ring with additive…

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Q: Let R be a ring and a, be R. Show that ab is nilpotent implies that ba is nilpotent.

A: Solution. Since ab is nilpotent, (ab)n = 0 for some n ∈ N.

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: Let R be a commutative ring with identity, and let a, b E R. Assume ab is a unit in R. Do a and b…

A: Here given R is a commutative ring with identity. and let a,b∈R and assume ab is a unit. we have to…

Q: Let R be a ring. Let p= ax + b and q = cx² + dx + e be elements of R[x], for a, b, c, d, e E R. True…

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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: Given information : Let R be a commutative ring, a,b∈R and abis a zero divisor.

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: ve integers. Show that the quotient ring ne. nZ nmZ has a multiplicative ide

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Q: Show that a ring R is commutative if and only it a - b = (a+ b) (a - b) for all a, be R.

A: Proof. Let R be commutative. Then ab = ba for all a,b ∈ R.

Q: Let R be a ring, and suppose that 0 = 1 in R. Show that Ris the zero ring. In other words: show that…

A: Consider the ring  with  in .To show that  is the zero ring.That is to show that if  is any element…

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…

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