Why 2Z has no multiplicative identity

Detailed Solution

$2\mathbb Z$ is a commutative ring, inheriting associativity, commutativity and distributivity from $\mathbb Z$, so (A) is true. For $2\mathbb Z$ to have a multiplicative identity $e=2k$, we would need $(2k)(2m)=2m$ for every $m\in\mathbb Z$, i.e. $4km=2m$ for all $m$, forcing $k=1/2$ — not an integer. So no element of $2\mathbb Z$ can serve as unity, confirming (B), and $1\notin2\mathbb Z$ directly confirms (C). Option (D) is the false statement: subrings do not automatically inherit the unity of their parent ring, and $2\mathbb Z$ is precisely a classic counterexample — a "rng" (ring without unity) sitting inside a ring that does have one. Thus the correct option is $\boxed{\mathrm{D}}$.