# nLab subring

Contents

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Idea

In algebra, a subring is a subobject of a ring, i.e. a subobject of an object in the category Ring of rings with homomorphisms between them.

As usual, there is a little bit of variation of what exactly one takes to be the definition of “ring” (multiplicative unitality is usually understood by default, while commutativity is usually not assumed by default) but in each case the general notion of subobject reduces to the appropriare notion of subring. For instance, a subring in the category of unital rings necessarily contains the unit-element of the ambient ring, etc.