The Jacobson radical of a ring has various internal and external characterizations.
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See the article " radical of a ring " for more of this.
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The only idempotent contained in the Jacobson radical of a ring is 0.
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In mathematics, specifically ring theory, the notion of "'quasiregularity "'provides a computationally convenient way to work with the Jacobson radical of a ring.
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The concept of the Jacobson radical of a ring; that is, the intersection of all right / left simple right / left modules over a ring, is one example.
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As a ring, it is sometimes a field, and often a direct product of fields; it can, though, contain non-zero nilpotents ( see radical of a ring ).
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In yet a simpler way, we may think of the Jacobson radical of a ring as method to " mod out bad elements " of the ring that is, members of the Jacobson radical act as 0 in the quotient ring, " R " / " J " ( " R " ).
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The Jacobson radical of a ring consists of elements that satisfy a stronger property than being merely a non-unit in some sense, a member of the Jacobson radical must not " act as a unit " in " any " 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.
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