Category Theoretic Description For Commutative Rings
When R is commutative and M is an R-module, we may describe AnnR(M) as the kernel of the action map R→EndR(M) determined by the adjunct map of the identity M→M along the Hom-tensor adjunction.
More generally, given a bilinear map of modules, the annihilator of a subset is the set of all elements in that annihilate :
Conversely, given, one can define an annihilator as a subset of .
The annihilator gives a Galois connection between subsets of and, and the associated closure operator is stronger than the span. In particular:
- annihilators are submodules
An important special case is in the presence of a nondegenerate form on a vector space, particularly an inner product: then the annihilator associated to the map is called the orthogonal complement.
Read more about this topic: Annihilator (ring Theory)
Famous quotes containing the words category, description and/or rings:
“The truth is, no matter how trying they become, babies two and under don’t have the ability to make moral choices, so they can’t be “bad.” That category only exists in the adult mind.”
—Anne Cassidy (20th century)
“As they are not seen on their way down the streams, it is thought by fishermen that they never return, but waste away and die, clinging to rocks and stumps of trees for an indefinite period; a tragic feature in the scenery of the river bottoms worthy to be remembered with Shakespeare’s description of the sea-floor.”
—Henry David Thoreau (1817–1862)
“You held my hand
and were instant to explain
the three rings of danger.”
—Anne Sexton (1928–1974)