The Earlier and the New Versions of the Definitions of Commutative Rings, Fields and Vector Spaces Agree

In the setting of Routine Facts, Continued: Typed Arithmetic, Maps and Predicativity.

(i) Commutative rings. For every set $r$, all binary operations $+$ and $\cdot$ on $r$ and all $0,1\in r$, the set $r$, with $+$, $\cdot$, $0$ and $1$, is a commutative ring in the earlier version if and only if it is a commutative ring in the new version.

(ii) Fields. For $r$, $+$, $\cdot$, $0$ and $1$ as in (i), $r$ is a field in the earlier version if and only if it is a field in the new version.

(iii) Vector spaces. For every field $K$ (in the new version), with $+$, $\cdot$, $0$ and $1$, every set $V$, every binary operation $+$ on $V$ and every function from $K\times V$ to $V$, the set $V$, with this addition and scalar multiplication, is a vector space over $K$ in the earlier version if and only if it is a vector space over $K$ in the new version; and $V$ is a real vector space in the earlier version if and only if it is a real vector space in the new version.