Question

Q. What is sub-ring? Write its two example.

Answer 1

Answer on Question #62406 – Math – Abstract Algebra

Question

What is sub-ring? Write its two examples.

Solution

Definition

A subring $S$ of a ring $R$ is a subset of $R$ which is a ring under the same operations as $R$.

Equivalently:

The criterion for a subring

A non-empty subset $S$ of $R$ is a subring if $a, b \in S \Rightarrow a - b, ab \in S$.

So $S$ is closed under subtraction and multiplication.

Examples

1. The even integers $2\mathbb{Z}$ form a subring of $\mathbb{Z}$.

More generally, if $n$ is any integer the set of all multiples of $n$ is a subring $n\mathbb{Z}$ of $\mathbb{Z}$.

The odd integers do not form a subring of $\mathbb{Z}$.

2. The set $\{a + bi \in \mathbb{C} \mid a, b \in \mathbb{Z}\}$ forms a subring of $\mathbb{C}$.

This is called the ring of Gaussian integers (sometimes written $Z[i]$) and it is important in Number Theory.

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