f-ring

Remarks

Example: the ring $C(X)$ of continuous real functions on a topological space $X$, with the pointwise order, is an f-ring. If $\min(a, b) = 0$ pointwise and $c \ge 0$, then at each point where $b > 0$ one has $a = 0$, so $ca = 0$ there, and $\min(ca, b) = 0$. Non-example: the ring of real $2 \times 2$ matrices with the entrywise order is lattice-ordered but is not an f-ring. Let $E_{ij}$ be the matrix unit; then $E_{11} \wedge E_{21} = 0$ and $E_{21} \ge 0$, but $E_{21} E_{11} = E_{21}$ and $E_{21} \wedge E_{21} = E_{21} \ne 0$. Every totally ordered ring is an f-ring, and in an f-ring every square is nonnegative.