Let R be a commutative ring with unity and let S≠R be an ideal of R. Then R/S is an integral domain if and only if S is a prime ideal of R.
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Let R be a commutative ring with unity and let S≠R be an
ideal of R. Then R/S is an
S is a prime ideal of R.
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- 17. Suppose is a ring with positive characteristic. Prove that if is any ideal of then is a multiple of the characteristic of.Let R be a commutative ring that does not have a unity. For a fixed aR, prove that the set (a)={na+ra|n,rR} is an ideal of R that contains the element a. (This ideal is called the principal ideal of R that is generated by a. )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 .)
- If R is a finite commutative ring with unity, prove that every prime ideal of R is a maximal ideal of R.Exercises Let be an ideal of a ring , and let be a subring of . Prove that is an ideal of18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .