Below you will find example sentences with "jacobson radical". The examples show how this phrase is used in real sentences and which words often surround it.
Jacobson Radical in a sentence
About this phrase
- Belongs to the word: radical
Example types with jacobson radical
Below, the examples are grouped by length and sentence type:
The radical of a module extends the definition of the Jacobson radical to include modules. (15 words)
More generally: the Jacobson radical of every local ring is the unique maximal ideal of the ring. (17 words)
These notions are of course imprecise, but at least explain why the nilradical of a commutative ring is contained in the ring's Jacobson radical. (25 words)
More precisely, a member of the Jacobson radical must project under the canonical homomorphism to the zero of every "right division ring" (each non-zero element of which has a right inverse ) internal to the ring in question. (38 words)
The following are equivalent characterizations of the Jacobson radical in rings with unity (characterizations for rings without unity are given immediately afterward): * J(R) equals the intersection of all maximal right ideals of the ring. (35 words)
In fact for any ring, the nilpotent elements in the center of the ring are also in the Jacobson radical.sfn So, for commutative rings, the nilradical is contained in the Jacobson radical. (33 words)
Example sentences (8)
Elements of the Jacobson radical and nilradical can be therefore seen as generalizations of 0. Equivalent characterizations The Jacobson radical of a ring has various internal and external characterizations.
In fact for any ring, the nilpotent elements in the center of the ring are also in the Jacobson radical.sfn So, for commutative rings, the nilradical is contained in the Jacobson radical.
The radical of a module extends the definition of the Jacobson radical to include modules.
Isaacs, Corollary 13.4, p. 180 sfn Alternatively, one could replace "right" with "left" in the previous sentence.sfn This characterization of the Jacobson radical is useful both computationally and in aiding intuition.
More generally: the Jacobson radical of every local ring is the unique maximal ideal of the ring.
More precisely, a member of the Jacobson radical must project under the canonical homomorphism to the zero of every "right division ring" (each non-zero element of which has a right inverse ) internal to the ring in question.
The following are equivalent characterizations of the Jacobson radical in rings with unity (characterizations for rings without unity are given immediately afterward): * J(R) equals the intersection of all maximal right ideals of the ring.
These notions are of course imprecise, but at least explain why the nilradical of a commutative ring is contained in the ring's Jacobson radical.