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 I be an ideal in a ring R with unity. Prove that if I contains an element a that has a multiplicative inverse, then I=R.17. Suppose is a ring with positive characteristic. Prove that if is any ideal of then is a multiple of the characteristic of.a. If R is a commutative ring with unity, show that the characteristic of R[ x ] is the same as the characteristic of R. b. State the characteristic of Zn[ x ]. c. State the characteristic of Z[ x ].
- 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.If R is a finite commutative ring with unity, prove that every prime ideal of R is a maximal ideal of R.Let R be a commutative ring with unity whose only ideals are {0} and R Prove that R is a field.(Hint: See Exercise 30.)
- If R1 and R2 are subrings of the ring R, prove that R1R2 is a subring of R.21. Prove that if a ring has a finite number of elements, then the characteristic of is a positive integer.33. An element of a ring is called nilpotent if for some positive integer . Show that the set of all nilpotent elements in a commutative ring forms an ideal of . (This ideal is called the radical of .)