Exercise 13.5. Let R be a commutative ring and P be a proper ideal of R. Show that P is a prime ideal of R if and only if for every two ideal I and J of R, one has IJCP ICP or JC P.
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- If R is a finite commutative ring with unity, prove that every prime ideal of R is a maximal ideal of R.22. Let be a ring with finite number of elements. Show that the characteristic of divides .17. Suppose is a ring with positive characteristic. Prove that if is any ideal of then is a multiple of the characteristic of.
- 12. Let be a commutative ring with unity. If prove that is an ideal of.Exercises Find two ideals and of the ring such that is not an ideal of . is an ideal 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 ].