In abstract algebra, a partially ordered ring is a ring (A, +, ·), together with a compatible partial order, that is, a partial order ${\displaystyle \,\leq \,}$ on the underlying set A that is compatible with the ring operations in the sense that it satisfies: $${\displaystyle x\leq y{\text{ implies }}x+z\leq y+z}$$ and $${\displaystyle 0\leq x{\text{ and }}0\leq y{\text{ imply that }}0\leq x\cdot y}$$ for all ${\displaystyle x,y,z\in A}$. Various extensions of this definition exist that constrain the ring, the partial order, or both. For example, an Archimedean partially ordered ring is a partially ordered ring ${\displaystyle (A,\leq )}$ where ${\displaystyle A}$'s partially ordered additive group is Archimedean.

An ordered ring, also called a totally ordered ring, is a partially ordered ring ${\displaystyle (A,\leq )}$ where ${\displaystyle \,\leq \,}$ is additionally a total order.

An l-ring, or lattice-ordered ring, is a partially ordered ring ${\displaystyle (A,\leq )}$ where ${\displaystyle \,\leq \,}$ is additionally a lattice order.

The additive group of a partially ordered ring is always a partially ordered group.

The set of non-negative elements of a partially ordered ring (the set of elements ${\displaystyle x}$ for which ${\displaystyle 0\leq x,}$ also called the positive cone of the ring) is closed under addition and multiplication, that is, if ${\displaystyle P}$ is the set of non-negative elements of a partially ordered ring, then ${\displaystyle P+P\subseteq P}$ and ${\displaystyle P\cdot P\subseteq P.}$ Furthermore, ${\displaystyle P\cap (-P)=\{0\}.}$

The mapping of the compatible partial order on a ring ${\displaystyle A}$ to the set of its non-negative elements is one-to-one; that is, the compatible partial order uniquely determines the set of non-negative elements, and a set of elements uniquely determines the compatible partial order if one exists.

If ${\displaystyle S\subseteq A}$ is a subset of a ring ${\displaystyle A,}$ and:

then the relation ${\displaystyle \,\leq \,}$ where ${\displaystyle x\leq y}$ if and only if ${\displaystyle y-x\in S}$ defines a compatible partial order on ${\displaystyle A}$ (that is, ${\displaystyle (A,\leq )}$ is a partially ordered ring).