Definition:Vector Subspace/Proper Subspace

Definition
Let $K$ be a division ring.

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

Let $\struct {T, +_T, \circ_T}_K$ be a vector subspace of $\struct {S, +, \circ}_K$.

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