1Help me solve pic problem steps by steps As much as possibl

Solution

find the isomorphism rom R² to R²

The direct product of two rings, say R1 and R2, is written R1 x R2 and consists of all ordered pairs (a, b) with a in R1 and b in R2. The addition rule for such pairs is (a, b) + (c, d) = (a+c, b+d) and the multiplication rule is (a,b) (c,d) = (ac, bd). If R1 and R2 are rings, then R1 x R2 is also a ring, as you could show by checking that all the axioms hold.

Take as a concrete example R1 = Z2 and R2 = Z3. Then Z2 x Z3 has as elements the 6 pairs 00, 01, 02, 10, 11, 12. (We write them here without the brackets or commas to save typing.) We can easily construct the Cayley + table:

and the Cayley x table:

Now compare these tables with the Cayley + and x tables for Z6:

You should be able to check that both + and both x tables are identical, provided you identify

00 with 0, so phi(0) = 00
11 with 1, so phi(0) = 11
02 with 2, so phi(0) = 02
10 with 3, so phi(0) = 10
01 with 4, so phi(0) = 01
12 with 5, so phi(0) = 12

This re-labelling is just a mapping phi : Z2 x Z3 -> Z6, which is one-to-one and onto and so is an isomorphism. Note that, in this case, phi is a discrete mapping.

Hence, Z2 x Z3 is isomorphic to Z6.