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

The new versions Commutative Rings, Fields and Vector Space over a Field have the texts of the earlier versions Commutative Rings, Fields and Vector Space over a Field, with the same settings; the only change is that the new definitions of fields and of vector spaces refer to the new definitions of commutative rings and of fields. The data introduced in the opening sentences are therefore the same in both versions, and it remains to compare the defining conditions clause by clause.

For commutative rings, Commutative Rings (i) and Commutative Rings (i) impose on $r$, $+$, $\cdot$, $0$ and $1$ the same seven displayed identities and the same existence of additive inverses. This proves (i).

For fields, Fields (i) and Fields (i) both require the commutative ring condition, which is the same in the two versions by (i), together with $0\neq1$ and the existence, for every $x\neq0$, of a $y$ with $x\cdot y=1$. This proves (ii).

For vector spaces, let $K$ be a field in the new version; by (ii) it is a field in the earlier version, so the earlier definition applies to it. Then Vector Space over a Field (ix) and Vector Space over a Field (ix) both require the same eight conditions, Vector Space over a Field (i) to Vector Space over a Field (viii) and Vector Space over a Field (i) to Vector Space over a Field (viii), on the same addition and scalar multiplication over the same field, the field condition being the same in both versions by (ii). The real vector spaces of Vector Space over a Field (x) and Vector Space over a Field (x) are in both versions the vector spaces over $\mathbb{R}$ with its operations and constants. This proves (iii).