What are ideals?

Some articles on ideals, ideal:

Manhauling - Modern Perspective - Ideals
... Similar to the modern, fast, free, and equipment light Alpine climbing ideals, modern manhaul purists seek to achieve mileage on their journey where possible ...
Minimal Prime (commutative Algebra) - Properties
... Every proper ideal I in a commutative ring has at least one minimal prime ideal above it ... Any maximal ideal containing I is prime, and such ideals exist, so the set of prime ideals containing I is non-empty ... of a decreasing chain of prime ideals is prime ...
Subring Test - Relation To Ideals
... Proper ideals are subrings that are closed under both left and right multiplication by elements from R ... and otherwise conform to the ring structure, and ideals become subrings ... Ideals may or may not have their own multiplicative identity (distinct from the identity of the ring) The ideal I = {(z,0)

Famous quotes containing the word ideals:

    With the breakdown of the traditional institutions which convey values, more of the burdens and responsibility for transmitting values fall upon parental shoulders, and it is getting harder all the time both to embody the virtues we hope to teach our children and to find for ourselves the ideals and values that will give our own lives purpose and direction.
    Neil Kurshan (20th century)

    My own ideals for the university are those of a genuine democracy and serious scholarship. These two, indeed, seem to go together.
    Woodrow Wilson (1856–1924)

    We want our children to become warm, decent human beings who reach out generously to those in need. We hope they find values and ideals to give their lives purpose so they contribute to the world and make it a better place because they have lived in it. Intelligence, success, and high achievement are worthy goals, but they mean nothing if our children are not basically kind and loving people.
    Neil Kurshan (20th century)