In a vector space $V$ over a field $F$, a nonempty subset $W$ of $V$ is a subspace if it is closed under addition and scalar multiplication. For a module $M$ over a ring $R$ with identity the similar result is true. Is it true in modules over a nonunital ring?

I should mention that the analog analogue of the following is not true.

In a vector space $V$ over a field $F$, a nonempty subset $W$ of $V$ is a subspace if for all scalars $\alpha_1, \alpha_2$ and for all $w_1, w_2 \in W$, $\alpha_1 w_1 + \alpha_2 w_2 \in W$.

This can be seen by the module $Z$ \mathbb{Z}$ over $2Z$ 2\mathbb{Z}$ and the subset $2Z 2\mathbb{Z} \cup {-3,3}.

Love

Dinesh Karia \{-3,3\}$.

In a vector space $V$ over a field $F$, a nonempty subset $W$ of $V$ is a subspace if it is closed under addition and scalar multiplication. For a module $M$ over a ring $R$ with identity the similar result is true. Is it true in modules over a nonunital ring?

I should mention that the analog of the following is not true.

In a vector space $V$ over a field $F$, a nonempty subset $W$ of $V$ is a subspace if for all scalars $\alpha_1, \alpha_2$ and for all $w_1, w_2 \in W$, $\alpha_1 w_1 + \alpha_2 w_2 \in W$.

This can be seen by the module $Z$ over $2Z$ and the subset $2Z \cup {-3,3}.

Love

Dinesh Karia