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:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)