Definitions
For an arbitrary ring, let be the underlying additive group. A subset is called a two-sided ideal (or simply an ideal) of if it is an additive subgroup of R that "absorbs multiplication by elements of R". Formally we mean that is an ideal if it satisfies the following conditions:
- is a subgroup of
Equivalently, an ideal of R is a sub-R-bimodule of R.
A subset of is called a right ideal of if it is an additive subgroup of R and absorbs multiplication on the right, that is:
- is a subgroup of
Equivalently, a right ideal of is a right -submodule of .
Similarly a subset of is called a left ideal of if it is an additive subgroup of R absorbing multiplication on the left:
- is a subgroup of
Equivalently, a left ideal of is a left -submodule of .
In all cases, the first condition can be replaced by the following well-known criterion that ensures a nonempty subset of a group is a subgroup:
- 1. is non-empty and .
The left ideals in R are exactly the right ideals in the opposite ring Ro and vice versa. A two-sided ideal is a left ideal that is also a right ideal, and is often called an ideal except to emphasize that there might exist single-sided ideals. When R is a commutative ring, the definitions of left, right, and two-sided ideal coincide, and the term ideal is used alone.
Read more about this topic: Ideal (ring Theory)
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