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### Why in Quotient Group we need normal subgroup? [duplicate]

For a group $G$ and a normal subgroup $N$ of $G$, the quotient group of $N$ in $G$, written $G/N$ and read "$G$ modulo $N$", is the set of cosets of $N$ in $G$. Question : Why in the defintion of ...
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### Associativity, commutativity and distributivity of modulo arithmetic

Textbooks usually state "it is not hard to check that in modular arithmetic the usual associative, commutative and distributive properties continue to apply". Is there a way other than tedious proof ...
### Find all subrings of $\mathbb{Z}^2$
This may be a simple question: Find all subrings of $\mathbb{Z}^2$.
One way to prove that a field $K$ has no ideals except the entire field and the trivial ideal is to note the fact that every element $x$ has an inverse. By the definition of an ideal, if $x$ is in the ...