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?


A ring has a group structure for its addition. This means that viewing a ring $R$ with just its addition and ignoring the multiplication will leave just a group where $+$ is the group operation. So Yes,

$$xH\qquad\text{means}\qquad x+H$$

in the ring.


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