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Answer to Question #4998 in Abstract Algebra for Annemarie

Question #4998
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?
Expert's answer
the elements of Z2is 0 and 1 let denote it by 0Z2 and 1Z2.
the elements of Z3is 0, 1 and 2 let denote it by 0Z3, 1Z3.,2Z3
a) elements of Z2⊕ Z3 are the pairs
(0Z2 ,0Z3), (0Z2 , 1Z3), (0Z2 , 2Z3),(1Z2 , 0Z3), (1Z2 , 1Z3), (1Z2, 2Z3)
elements of Z3⊕ Z2 are the pairs
(0Z3 ,0Z2), (0Z3 , 1Z2), (1Z3 , 0Z2),(1Z3 , 1Z2), (2Z3 , 0Z2), (2Z3, 1Z2)
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

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