Category of rings with arbitrary ring homomorphisms

The category of rings with arbitrary ring homomorphisms has associative rings as objects, without imposing a multiplicative identity. A morphism $f:R\to S$ preserves addition and multiplication: $f(x+y)=f(x)+f(y)$ and $f(xy)=f(x)f(y)$ . If units happen to exist, this category does not require $f(1_R)=1_S$ . Identities and composition are ordinary functions.

Convention and comparison

The zero map is always allowed. For unital objects and maps preserving their units, use the category of unital rings. The distinction is stored explicitly in the catalogue.