About definition of product over an additive group

Question

We see that on the cyclic additive group $(\mathbb Z,+) $ we can define a "natural" product induced by the sum. Let $(G,+) $ be a group, I wonder if there exists such a way I can define a function $\cdot : G\times G \rightarrow G $ such that $(G,+,\cdot ) $ is a ring (and the product is not the null function $a\cdot b=0\ \forall a,b\in G $). Or also, are there any conditions the group G has to satisfy in order to be given the structure of ring?