A apple2010 Mar 2010 8 0 May 12, 2010 #1 Let R be a nonzero ring. Prove that the following are equivalent: a)R is a field b)The only ideals in R are (0) and (1). c)Every homomorphism of R into a nonzero ring S is injective.

Let R be a nonzero ring. Prove that the following are equivalent: a)R is a field b)The only ideals in R are (0) and (1). c)Every homomorphism of R into a nonzero ring S is injective.

N NonCommAlg MHF Hall of Honor May 2008 2,295 1,663 May 12, 2010 #2 apple2010 said: Let R be a nonzero ring. Prove that the following are equivalent: a) R is a field b) The only ideals in R are (0) and (1). c) Every homomorphism of R into a nonzero ring S is injective. Click to expand... you forgot to mention that R is commutative. a) --> b) is trivial by the definition of a field. for b) --> c) look at the kernel of the homomorphism, which is an ideal of R. for c) --> a) suppose that r is a non-zero element of R and put I = Rr, S = R/I and take f : R ---> S to be the natural homomorphism. Reactions: apple2010

apple2010 said: Let R be a nonzero ring. Prove that the following are equivalent: a) R is a field b) The only ideals in R are (0) and (1). c) Every homomorphism of R into a nonzero ring S is injective. Click to expand... you forgot to mention that R is commutative. a) --> b) is trivial by the definition of a field. for b) --> c) look at the kernel of the homomorphism, which is an ideal of R. for c) --> a) suppose that r is a non-zero element of R and put I = Rr, S = R/I and take f : R ---> S to be the natural homomorphism.