Question

If
R1, ..., Rn
are commutative rings, define their direct product by induction on ( it is the set of n- tuples ( $r_{1,}....r_{n}$ ) with for all i). Prove that the ring $(\mathbb{Z})^{X}$ where is the set with is the direct product of copies of $\mathbb{Z}_{2}$ .

Answer 1

n times. We will make an identification map between (3)x and TXIX ...x2 Let, &: 2 > ZXZX...x2 is defined by for, fert, & (f)= ( f (oc), f), ..., f (x)) Where, x= {x, , x , ,--, an} So, if f f g E7X then f (x;) & g (x;) for some ithen? So, & (f) & 0 (9) and hence & is injective For any (an, ang-, an) E ZLXZX--XIL, define fEzLX f (x) =a , f (x2) = Az anf (an) = an. So, & (f) = (a... An) and hence & is sujective- So, & is an identification and 7 X LX X X n time ntimes