18.6.2. Let R be a local commutative ring with identity, and assume that the maximal ideal in R is a principal ideal. Let a be an irreducible element of R. Show that (a) is maximal.
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- 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 ].If R1 and R2 are subrings of the ring R, prove that R1R2 is a subring of R.. a. Let, and . Show that and are only ideals of and hence is a maximal ideal. b. Show that is not a field. Hence Theorem is not true if the condition that is commutative is removed. Theorem 6.22 Quotient Rings That are Fields. Let be a commutative ring with unity, and let be an ideal of . Then is a field if and only if is a maximal ideal of .