A subring is a subset of a ring that is also a ring itself. It contains the same operations (addition and multiplication) and follows the same rules as the larger ring.
To determine if S is a subring of R, you need to check if S satisfies the three subring conditions: closure under addition and multiplication, contains the identity element, and contains the additive and multiplicative inverses of each element.
If S is a proper subring of R, it means that S is a subring of R, but not equal to R. This means that S contains a subset of the elements in R and follows the same operations and rules, but is not the same as R.
No, a subring must have the same identity element as the larger ring. This is one of the three subring conditions that must be satisfied in order for S to be a subring of R.
A subring is a subset of a ring that is also a ring, while a subgroup is a subset of a group that is also a group. While both subrings and subgroups follow the same operations and rules as the larger structure, they differ in the type of structure they belong to (ring vs. group).