Question #109443
Expert's answer
Let's consider the following structure of R5, R4 and R:
R5 = {(r1, r2, r3, r4, r5) | ri ∈ R for i = 1,2,3,4,5},
R4 = {(r1, r2, r3, r4) | ri ∈ R for i= 1,2,3,4},
R = {(0, 0, 0, 0, r) | r ∈ R }, with ring operations applied to each component(i.e. (r1, r2, r3, r4, r5) +(or *) (p1, p2, p3, p4, p5) = (r1+(*)p1, ..., r5+(*)p5) and same for R4 and R).
It could be inferred(from the expressions, represented above) that R5/R = {(r1, r2, r3, r4, 0) | ri ∈ R for i = 1, 2, 3, 4}. Here is the isomorphism between the R5/R and R4:
f(r1, r2, r3, r4, 0) = (r1, r2, r3, r4).
f-1(r1, r2, r3, r4) = (r1, r2, r3, r4, 0).
Let's check f and f-1 are homomorphisms of rings:
Since the last component of the element from R5/R always equals to zero(and due to our definition of ring operations) f is a homomorphism:
f(a + b) = f(a) + f(b)
f(a*b) = f(a)*f(b) where a,b belong to the R5/R
f-1 = g is just adding the 0 as the last componet. So, because of the definition of ring operations =>
g(a + b) = g(a) + g(b)
g(a*b) = g(a)*g(b) where a,b belong to the R4
This means, that f is invertible homomorphism => R5/R ~f R4.
