I'm reading "A First Course in Abstract Algebra, John B. Fraleigh, 7th edition", and I'm having trouble seeing the connection between the following two theorems:
14.9 Theorem Let $H$ be a normal subgroup of $G$. Then $\gamma: G \to G/H$ given by $\gamma(x) = xH$ is a homomorphism with kernel $H$.
26.16 Theorem (Analogue of Theorem 14.9) Let $N$ be an ideal of a ring $R$. Then $\gamma: R \to R/N$ given by $\gamma(x) = x + N$ is a ring homomorphism with kernel $N$.
Now, in the proof of Theorem 26.16 he prefaces by saying:
The additive part is done in Theorem 14.9
So my question is this, does $xH$ in Theorem 14.9 really denote $x + H$ where the operation is implicit, or are they different expressions all together?