Download A First Course in Rings and Ideals by David M. Burton PDF

Zn by takingf(a) = [a]; that is, rnap each integer into the congruence class containing it.

The next theorem indicates the algebraic nature of direct and inverse images of subrings under homomorphisms. Among other things, we shall see that iffis a homomorphism from the ring R into the ring R', thenf(R) forms a subring of R'. The complete story is told below. Theorem 2-8. Letfbe a homomorphism from the ring R intotqe ring R'. Then, ... " 1) for each subring S of R,J(S) is a subring of R'; a n d ' . l(S') is a subring of R. = Ta(J)~a(g). l. We now list sorne of the structural features preserved under hornornorphisrns.

This in itself would amply justify the study of such rings. 1 = (filfE R;f(O) = O}, J = {ji2 + ni21fE R;f(O) = O; n E Z}, We now turn our attention to functions between rings arid, more specifically, to functions which preserve both the ring operations. l: r ~,J¡ = fa E Rla(~ J;)~ 1} = {q:E RlaJ¡' ~ Ifor all i} n (1 :r J¡). Confirmation of the final' assertion follows from l: r (JK) = {a~,Rla(JK) ~J} = {aERI(aJ)K ~ 1}' = {~ERlaJ ~ l: r K} = (1 :r K) :r J. where i denotes the identity function on R # (that is, i(x) = x for all x E R #).

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