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It is given that $X=\{0,1,2,3\}$ forms a group under addition modulo $4$ and now we have to find the number of possible ways a multiplication can be defined on $X$ so as to make it a ring under $+_4$ and $\times$

Now there can be 16 different possible products with the given elements of $X$. Again If $X$ has to be closed under the multiplication then there are only $4$ possible values for each of the products therefore total number of ways we can define the multiplication is $4^{16}$. Am I right in this? How do I bring in the condition of distribution of the multiplication over addition?

[I am not yet clear with the answers that has been provided by Monstah. Further clarification will be highly appreciated].

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Actually, since being a ring requires product distribuctivity in relation to the sum, you only need to define the products for the generators, and the rest will follow by linearity.

In your case, X has only 1 as generator, so your choice for 1*1 determines the whole product table.

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  • $\begingroup$ And what that number will be? It should be only 4 then right? @ Monstah $\endgroup$
    – Sourav Chakraborty
    Commented Dec 20, 2013 at 18:48
  • $\begingroup$ So does this mean that rest all i.e 4$^{16}$ -4 other possible ways of defining the multiplication will not form a ring ? $\endgroup$
    – Sourav Chakraborty
    Commented Dec 20, 2013 at 18:54
  • $\begingroup$ Sorry it took long to reply. Yes, you are correct, in this case the number of options is 4. $\endgroup$
    – Monstah
    Commented May 9, 2014 at 13:41

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