Checking the ring axioms

(a) $(\mathbb{Z},+)$ abelian group, multiplication associative and distributive. Unit $1$. Commutative ring with unit.

(b) $2\mathbb{Z}$: abelian group, closed under multiplication ($2a\cdot2b=4ab\in2\mathbb{Z}$), associative and distributive. But no unit (1 is not in $2\mathbb{Z}$). Commutative ring without unit (a pseudo-ring).

(c) $(\mathbb{N},+)$ is not a group (no additive inverses). Not a ring.

(d) $\mathbb{R}[X]$: polynomial addition is an abelian group, multiplication associative/distributive. Unit $1$. Commutative ring with unit.

(e) $M_n(\mathbb{R})$: abelian group under addition, matrix multiplication associative, distributive. Unit $I_n$. Non-commutative for $n\ge2$. Non-commutative ring with unit.

Conclusion. (a), (d), (e) are rings with unit; (b) is a ring without unit; (c) is not a ring.