Let R be a ring and assume a∈R is not a zero divisor.Prove that if ba=ca, then b=c.
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Q: if A, B, C are ideals of a ring R such that BCA, prove that An (B + C) = B+ (AnC) = (A B) + (AnC).
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Q: If A, B, C are ideals of a ring R such that BCA, prove that An (B + C) = B+ (AnC) = (An B) + (AnC).
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Q: Prove that if (I,+,.) is an ideal of the Ring (R,+,.) then rad I= In rad R ???
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Let R be a ring and assume a∈R is not a zero divisor.Prove that if ba=ca, then b=c.
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- Prove that if a is a unit in a ring R with unity, then a is not a zero divisor.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 .)37. Let and be elements in a ring. If is a zero divisor, prove that either or is a zero divisor.
- If R1 and R2 are subrings of the ring R, prove that R1R2 is a subring of R.14. Letbe a commutative ring with unity in which the cancellation law for multiplication holds. That is, if are elements of , then and always imply. Prove that is an integral domain.19. Find a specific example of two elements and in a ring such that and .
- 50. Let and be nilpotent elements that satisfy the following conditions in a commutative ring: Prove that is nilpotent. for some12. Let be a commutative ring with prime characteristic . Prove, for any in that for every positive integer .15. Prove that if is an ideal in a commutative ring with unity, then is an ideal in .