Consider the subset 5Z of the ring Z Weve seen that 5Z is an

Consider the subset 5Z of the ring, Z. We\'ve seen that 5Z is an additive subgroup of Z. Is 5Z a subring of Z? Why or why not? Show that 5Z satisfies the \"product absorption\" property, i.e., ri 5Z for all r Z and i 5Z

Solution

(a) The members of 5Z looks like {...-10,-5,0,5,10,15,...}

And so all the numbers are given by formula 5a, a is in Z

Now (i) 0 is in 5Z

(ii) Also 5a+5b = 5(a+b) in 5Z (because a,b in Z => a+b in Z)

(iii) 5a. 5b = 25 ab = 5(5ab) in 5Z (because a,b in Z => 5ab is in Z)

(iv) Now let 5a and 5b in 5Z such that a.b = 1 => 25ab = 1 => ab=1/25 => b = 1/25 a^-1

Let us say a=1 then b = 1/25 and that is not in integers

Hence 5Z has no multiplicative identity of Z so 5Z is not a subring of Z

(b) Let r in Z and i=5a in 5Z

Then ri = r(5a) = 5ra in 5Z (because ra is in Z)