Theorem assasca

Description: The scalars of an associative algebra form a ring. (Contributed by Mario Carneiro, 7-Jan-2015) (Revised by SN, 2-Mar-2025)

	Ref	Expression
Hypothesis	assasca.f	$$⊢F=\mathrm{Scalar}\left(W\right)$$
Assertion	assasca	$$⊢W\in \mathrm{AssAlg}\to F\in \mathrm{Ring}$$

Proof

Step	Hyp	Ref	Expression
1		assasca.f	$$⊢F=\mathrm{Scalar}\left(W\right)$$
2		assalmod	$$⊢W\in \mathrm{AssAlg}\to W\in \mathrm{LMod}$$
3	1	lmodring	$$⊢W\in \mathrm{LMod}\to F\in \mathrm{Ring}$$
4	2 3	syl	$$⊢W\in \mathrm{AssAlg}\to F\in \mathrm{Ring}$$