18.6.1. Let R be a non-trivial commutative ring with identity. Show that the following are equivalent: (a) R is local. (b) The set of all non-units of R forms an ideal of R. (c) The sum of any two non-units in R is a non-unit. (d) If x ЄR, then x or 1 - x is a unit.
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- Exercises If and are two ideals of the ring , prove that 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 ].15. Prove that if is an ideal in a commutative ring with unity, then is an ideal in .
- Exercises Find two ideals and of the ring such that is not an ideal of . is an ideal of .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.
- 19. Find a specific example of two elements and in a ring such that and .Assume that each of R and S is a commutative ring with unity and that :RS is an epimorphism from R to S. Let :R[ x ]S[ x ] be defined by, (a0+a1x++anxn)=(a0)+(a1)x++(an)xn Prove that is an epimorphism.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.