Defining $R\times R$ as a ring? I feel a bit stupid, but I know that the normal definition of $R\times R$ as $R \times S = \{(r, s) : r \in R, s \in S\}$, under $(r, s) + (r', s')=(r+r',s+s')$ and $(r, s) \cdot (r', s')=(rr', ss')$ is a ring.

But, can you define $R \times R$ otherwise as a ring?

I'm trying to decide whether $R \times R$ has any non-zero nilpotent elements. Obviously it does not under the normal definition, but can you define $R \times R$ as a ring otherwise such that there are non-zero nilpotent elements?

Yes, you can, but it is not necessarily useful. There is a very precise meaning to the symbol $\times$: it is the product in the category you are considering. It is, in some sense, the smaller object that contains the whole structure of both the object you are taking the product of (more precisely, it satisfies a certain universal property that you can find here). If you define the ring structure in another way, you lose this property.
