Theorem rngansg

Description: Every additive subgroup of a non-unital ring is normal. (Contributed by AV, 25-Feb-2025)

	Ref	Expression
Assertion	rngansg	$$⊢R\in \mathrm{Rng}\to \mathrm{NrmSGrp}\left(R\right)=\mathrm{SubGrp}\left(R\right)$$

Proof

Step	Hyp	Ref	Expression
1		rngabl	$$⊢R\in \mathrm{Rng}\to R\in \mathrm{Abel}$$
2		ablnsg	$$⊢R\in \mathrm{Abel}\to \mathrm{NrmSGrp}\left(R\right)=\mathrm{SubGrp}\left(R\right)$$
3	1 2	syl	$$⊢R\in \mathrm{Rng}\to \mathrm{NrmSGrp}\left(R\right)=\mathrm{SubGrp}\left(R\right)$$