> [!NOTE] **Definition** (Subring) > > Let $(R,+, \cdot)$ be a [[Rings|ring]]. Let $S$ be a [[Subsets|subset]] of $R$ and suppose that $(S,+, \cdot)$ is also ring (with the same unity if $R$ is a ring with unity). Then we say that $S$ is a subring of $R$ (or more formally, $(S,+, \cdot )$ is a subring of $(R, +, \cdot)$. # Properties **Subring Tests**: Let $S$ be a subset of a ring $R.$ Then $S$ is a subring of $R$ iff it passes the [[Three Step Subring Test|subring test]].