Question

Give the necessary and sufficient condition for a non-empty subset s of a ring (r, +, ) to be a subring of r.

Answer 1

Let (R,+,.) be a ring. A non empty subset S of R is called a subring of R if (S,+,.) is a ring.

For example the set {\displaystyle m\mathbb {Z} } {\displaystyle m\mathbb {Z} } which stands for {\displaystyle \{0,\pm m,\pm 2m\cdots \}} {\displaystyle \{0,\pm m,\pm 2m\cdots \}} is a subring of the ring of integers, the set of Gaussian integers {\displaystyle \mathbb {Z} [i]} {\displaystyle \mathbb {Z} [i]} is a subring of {\displaystyle \mathbb {C} } \mathbb {C} and the set {\displaystyle \mathbb {Z} _{4}} \mathbb {Z} _{4} has the set {\displaystyle \{{\overline {0}},{\overline {2}}\}} {\displaystyle \{{\overline {0}},{\overline {2}}\}} as a subring under addition and multiplication modulo 4.

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Answer 2

Answer:

There's a subring theorem, which states that a non-empty subset S of a ring R =(r,+,*) is a subring of R with two necessary and sufficient conditions.

Step-by-step explanation

The above statement will be true if and only if the following conditions hold:

1. S is closed under the  binary operation '+'
2. If ∈ S then a^-1 ∈ S , where a^-1 denotes the inverse of a under the binary operation +.

• A Subring of  R is a subset of a ring that is itself is a ring with binary operations of addition or multiplication.