Definition:Vector Subspace/Proper Subspace

Definition
Let $K$ be a division ring.

Let $\left({S, +, \circ}\right)_K$ be a $K$-algebraic structure with one operation.

Let $\left({T, +_T, \circ_T}\right)_K$ be a vector subspace of $\left({S, +, \circ}\right)_K$.

If $T$ is a proper subset of $S$, then $\left({T, +_T, \circ_T}\right)_K$ is a proper (vector) subspace of $\left({S, +, \circ}\right)_K$.