Given that S be a proper ideal of a ring R. Then our claim is that S is contained in a maximal 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. )36. Suppose that is a commutative ring with unity and that is an ideal of . Prove that the set of all such that for some positive integer is an ideal of .
- 15. Prove that if is an ideal in a commutative ring with unity, then is an ideal in .Label each of the following statements as either true or false. If I is an ideal of S where S is a subring of a ring R, then I is an ideal of R.21. Prove that if a ring has a finite number of elements, then the characteristic of is a positive integer.