Relations To Other Properties of Rings
- Annihilators are used to define left Rickart rings and Baer rings.
- The set of (left) zero divisors DS of S can be written as
(Here we allow zero to be a zero divisor.)
- In particular DS is the set of (left) zero divisors of R when S = R and R acts on itself as a left R-module.
- When R is commutative, the set DR is precisely equal to the union of the minimal prime ideals of R.
Read more about this topic: Annihilator (ring Theory)
Famous quotes containing the words relations to, relations, properties and/or rings:
“Subject the material world to the higher ends by understanding it in all its relations to daily life and action.”
—Ellen Henrietta Swallow Richards (1842–1911)
“So soon did we, wayfarers, begin to learn that man’s life is rounded with the same few facts, the same simple relations everywhere, and it is vain to travel to find it new.”
—Henry David Thoreau (1817–1862)
“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)
“Ye say they all have passed away,
That noble race and brave;
That their light canoes have vanished
From off the crested wave;
That, mid the forests where they roamed,
There rings no hunters’ shout;
But their name is on your waters,
Ye may not wash it out.”
—Lydia Huntley Sigourney (1791–1865)