Homological Algebra: R-homomorphism

Question

Please could you suggest what definitions I can use to demonstrate the following exercise.

For submodules $A, B$ of $R$-module left of $M$. Prove that $\dfrac{\left( A+B \right) }{B} \cong \dfrac{A}{\left( A \cap B \right) }$.

Answer 1

Define $f:\frac{A}{A\cap B}\to \frac{A+B}{B}$ by $$f:a+(A\cap B)\mapsto a+B,\forall a\in A.$$ Check that this is a well-defined $R$-module homomorphism. Also check that, $f$ is bijective.

1. Well-definedness:---

$a+(A\cap B)=a'+(a\cap B)\implies a-a'\in A\cap B\implies a-a'\in B\implies a+B=a'+B.$

2. $R$-module homomorphism :---

$f(r\cdot\overline a_1+\overline a_2)=f\big(\overline{ra_1+a_2}\big)=(ra_1+a_2)+B=\big(r\cdot (a_1+B)\big)+(a_2+B)=r\cdot f(\overline a_1)+f(\overline a_2).$

3. Injective :---

For $a\in A$ with $f(\overline a)=0+B\implies a+B=0+B\implies a\in B\implies a\in A\cap B.$

4. Surjective:---

For $(a+b)+B\in \frac{A+B}{B}$ we have $(a+b)+B=a+B=f(\overline a).$