Homework Sourse1Help me solve pic problem steps by steps As much as possibl
1.Help me solve pic problem steps by steps. As much as possible.
2.Also add a questions or conjecture after it ( for example, could we have finite order elements in an infinite group?)
2. How many functions are there from Z4 to Z4? How many of the functions are group isomorphisms? Solution
Let R and S be commutative rings, and let : R S be a ring homomorphism. (a) Does map idempotent elements to idempotent elements? Solution: Yes; if e 2 = e, then ((e))2 = (e 2 ) = (e). (b) Does map nilpotent elements to nilpotent elements? Solution: Yes; if x n = 0, then ((x))n = (x n ) = (0) = 0. (c) Does map zero divisors to zero divisors? Solution: No; let : Z2 Z2 Z2 be given by ((x, y)) = x. Then maps the zero divisor (1, 0) to 1, which is definitely not a zero divisor
