Let R be a ring without any nonzero nilpotent elements. Show that (ara - ra)² = 0 for all r E R and for all idempotent elements a E R. Hence, show that C(R) contains all idempotent elements.

Q: Show that the union of a chain I1 ⊂ I2 ⊂ .... of ideals of a ring R isan ideal of R.

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Q: Suppose that R is a commutative ring without zero-divisors. Showthat the characteristic of R is 0 or…

A: To Prove: if R is a commutative ring without zero-divisors then characteristic of R is 0 or prime.

Q: Suppose that a belongs to a ring and a4 = a2. Prove that a2n = a2 forall n >= 1.

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

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Q: 12. Let R² be the set of all pairs of real numbers R² = {(a,b)|a, b = R}. The addition is still…

A: is the set of all pairs of real numbers The multiplication is defined by

Q: Let R be a ring with unity. Assume b1a=1R and ab2=1R. Prove that b1=b2.   Also, let R be a ring…

A: Ring with unity,so 1R is different from 0R ,

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

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

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

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Q: Let R be a commutative ring of characteristic 2. Prove that : (a+ b) = a² +b² = (a - b)? v a, be R.…

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

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Q: III. An element x of a ring R is called idempotent if x² = x. Determine all the idempotent elements…

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Q: The population of a community is known to increase at a rate proportional to the number of people…

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Q: Suppose that R₁ is a ring with addition +1 and multiplication 1, and that R2 is a ring with addition…

A: Answer in steps:We need to show that (R₁ × R₂, +, ·) satisfies all the axioms for a ring.i.Closure…