If " x " is in " R ", then Rx and xR are left ideals and right ideals, respectively; they are called the principal left ideals and right ideals generated by " x ".
32.
The fact that the Jacobson radical can be viewed as the intersection of all maximal right / left ideals in the ring, shows how the internal structure of the ring is reflected by its modules.
33.
As of the 1980s, the theoretical physics group at Birkbeck College around David Bohm and Basil Hiley has been developing algebraic approaches to quantum theory that build on Sauter and Riesz'identification of spinors with minimal left ideals.
34.
A ring is "'left Artinian "'if it satisfies the descending chain condition on left ideals, "'right Artinian "'if it satisfies the descending chain condition on right ideals, and "'Artinian "'or "'two-sided Artinian "'if it is both left and right Artinian.
35.
It is also a fact that the intersection of all maximal right ideals in a ring is the same as the intersection of all maximal left ideals in the ring, in the context of all rings; whether commutative or noncommutative.
36.
In noncommutative ring theory, a "'maximal right ideal "'is defined analogously as being a maximal element in the poset of proper right ideals, and similarly, a "'maximal left ideal "'is defined to be a maximal element of the poset of proper left ideals.
37.
In noncommutative ring theory, a "'maximal right ideal "'is defined analogously as being a maximal element in the poset of proper right ideals, and similarly, a "'maximal left ideal "'is defined to be a maximal element of the poset of proper left ideals.
38.
K�the's paper also required a uniserial ring to have a unique composition series, which not only forces the right and left ideals to be linearly ordered, but also requires that there be only finitely many ideals in the chains of left and right ideals.
39.
;"'Noetherian ring "': A ring satisfying the ascending chain condition for left ideals is " left Noetherian "; a ring satisfying the ascending chain condition for right ideals is " right Noetherian "; a ring that is both left and right Noetherian is " Noetherian ".
40.
;"'artinian ring "': A ring satisfying the descending chain condition for left ideals is " left artinian "; if it satisfies the descending chain condition for right ideals, it is " right artinian "; if it is both left and right artinian, it is called " artinian ".
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