18.1.2. Let R be a commutative ring with identity, and let a, b E R. Assume ab is a unit in R. Do a and b have to be units?
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- 15. Let and be elements of a ring. Prove that the equation has a unique solution.Exercises If and are two ideals of the ring , prove that the set is an ideal of that contains each of and . The ideal is called the sum of ideals of and .Let R be a commutative ring with characteristic 2. Show that each of the following is true for all x,yR a. (x+y)2=x2+y2 b. (x+y)4=x4+y4
- 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 .)Let R and S be arbitrary rings. In the Cartesian product RS of R and S, define (r,s)=(r,s) if and only if r=r and s=s, (r1,s1)+(r2,s2)=(r1+r2,s1+s2), (r1,s1)(r2,s2)=(r1r2,s1s2). Prove that the Cartesian product is a ring with respect to these operations. It is called the direct sum of R and S and is denoted by RS. Prove that RS is commutative if both R and S are commutative. Prove RS has a unity element if both R and S have unity elements. Given as example of rings R and S such that RS does not have a unity element.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 ].
- Examples 5 and 6 of Section 5.1 showed that P(U) is a commutative ring with unity. In Exercises 4 and 5, let U={a,b}. Is P(U) a field? If not, find all nonzero elements that do not have multiplicative inverses. [Type here][Type here]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.18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .