Let R be a ring with unity and assume a ∈ R is a unit. Prove that a is not nilpotent.
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A: SOLUTION: The set of all units of the ring Z8={0,2,4,6} because, f(0)=f(2)=f(4)=f(6)=0
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A: Solution. Since ab is nilpotent, (ab)n = 0 for some n ∈ N.
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A: Explanation of the answer is as follows
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Q: Let R be a commutative ring with 1 ≠ 0. Prove that R is a field if and only if 0 is a maximal ideal.
A: We are given that R be a commutative ring with unity. We have to show that R is a field if and only…
Let R be a ring with unity and assume a ∈ R is a unit. Prove that a is not nilpotent.
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- 24. If is a commutative ring and is a fixed element of prove that the setis an ideal of . (The set is called the annihilator of in the ring .)Exercises Let be an ideal of a ring , and let be a subring of . Prove that is an ideal ofIf R1 and R2 are subrings of the ring R, prove that R1R2 is a subring of R.
- Let I be the set of all elements of a ring R that have finite additive order. Prove that I is an ideal of R.Assume R is a ring with unity e. Prove Theorem 5.8: If aR has a multiplicative inverse, the multiplicative inverse of a is unique.19. Find a specific example of two elements and in a ring such that and .