Answer to Question #4998 in Abstract Algebra for Annemarie

Question

Let R and S be rings. The direct product of R and S,
R ⊕ S = {(r, s) : r ∈ R, s ∈ S}
is a ring, where
(r1, s1) + (r2, s2) = (r1 + r2, s1 + s2)
(r1, s1) x (r2, s2) = (r1 x r2, s1 x s2)

(a) List the elements of Z2 ⊕ Z3 and Z3 ⊕ Z2.
(b) Are they structurally the same (that is isomorphic)? If so, how should an
element in Z2 ⊕ Z3 be identified in Z3 ⊕ Z2?

Answer 1

the elements of Z₂is 0 and 1 let denote it by 0_Z2 and 1_Z2.
the elements of Z₃is 0, 1 and 2 let denote it by 0_Z3, 1_Z3.,2_Z3
a) elements of Z₂⊕ Z₃ are the pairs
(0_Z2 ,0_Z3), (0_Z2 , 1_Z3), (0_Z2 , 2_Z3),(1_Z2 , 0_Z3), (1_Z2 , 1_Z3), (1_Z2, 2_Z3)
elements of Z₃⊕ Z₂ are the pairs
(0_Z3 ,0_Z2), (0_Z3 , 1_Z2), (1_Z3 , 0_Z2),(1_Z3 , 1_Z2), (2_Z3 , 0_Z2), (2_Z3, 1_Z2)
b) this rings are isomorphic because we can show function
f: Z₂⊕ Z₃ -->

Z₃⊕ Z₂ that permute coordinates
for x from Z₂and y from Z₃ f((x,y))=(y,x).
this function is a injection and surjection and it preserve operations:
for x,z from Z₂and y,t from Z₃
f((x,y)+(z,t))=f((x+z,y+t))=(y+t,x+z)=(y,x)+(t,z)=f((x,y))+f((z,t)).
f((x,y)*(z,t))=f((x*z,y*t))=(y*t,x*z)=(y,x)*(t,z)=f((x,y))*f((z,t))

So we have that rings isomorphic