Properties
{0} and R are ideals in every ring R. If R is a division ring or a field, then these are its only ideals. The ideal R is called the unit ideal. I is a proper ideal if it is a proper subset of R, that is, I does not equal R.
Just as normal subgroups of groups are kernels of group homomorphisms, ideals have interpretations as kernels. For a nonempty subset A of R:
- A is an ideal of R if and only if it is a kernel of a ring homomorphism from R.
- A is a right ideal of R if and only if it is a kernel of a homomorphism from the right R module RR to another right R module.
- A is a left ideal of R if and only if it is a kernel of a homomorphism from the left R module RR to another left R module.
If p is in R, then pR is a right ideal and Rp is a left ideal of R. These are called, respectively, the principal right and left ideals generated by p. To remember which is which, note that right ideals are stable under right-multiplication (IR ⊆ I) and left ideals are stable under left-multiplication (RI ⊆ I).
The connection between cosets and ideals can be seen by switching the operation from "multiplication" to "addition".
Read more about this topic: Ideal (ring Theory)
Famous quotes containing the word properties:
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—John Locke (1632–1704)
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