Semiprimitive ring - Misplaced Pages

Algebraic structure → Ring theory Ring theory
Basic conceptsRings • Subrings • Ideal • Quotient ring • Fractional ideal • Total ring of fractions • Product of rings • Free product of associative algebras • Tensor product of algebras Ring homomorphisms • Kernel • Inner automorphism • Frobenius endomorphism Algebraic structures • Module • Associative algebra • Graded ring • Involutive ring • Category of rings • Initial ring $\mathbb {Z}$ • Terminal ring $0=\mathbb {Z} /1\mathbb {Z}$ Related structures • Field • Finite field • Non-associative ring • Lie ring • Jordan ring • Semiring • Semifield
Commutative algebraCommutative rings • Integral domain • Integrally closed domain • GCD domain • Unique factorization domain • Principal ideal domain • Euclidean domain • Field • Finite field • Polynomial ring • Formal power series ring Algebraic number theory • Algebraic number field • Integers modulo n • Ring of integers • p-adic integers $\mathbb {Z} _{p}$ • p-adic numbers $\mathbb {Q} _{p}$ • Prüfer p-ring $\mathbb {Z} (p^{\infty })$
Noncommutative algebraNoncommutative rings • Division ring • Semiprimitive ring • Simple ring • Commutator Noncommutative algebraic geometry Free algebra Clifford algebra • Geometric algebra Operator algebra
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Algebraic structure → Ring theory
Ring theory

Basic conceptsRings

• Subrings

• Ideal

• Quotient ring

• Fractional ideal

• Total ring of fractions

• Product of rings

• Free product of associative algebras

• Tensor product of algebras

Ring homomorphisms

• Kernel

• Inner automorphism

• Frobenius endomorphism

Algebraic structures

• Module

• Associative algebra

• Graded ring

• Involutive ring

• Category of rings

• Initial ring

\mathbb {Z}

• Terminal ring

0=\mathbb {Z} /1\mathbb {Z}

Related structures

• Field

• Finite field

• Non-associative ring

• Lie ring

• Jordan ring

• Semiring

• Semifield

Commutative algebraCommutative rings

• Integral domain

• Integrally closed domain

• GCD domain

• Unique factorization domain

• Principal ideal domain

• Euclidean domain

• Field

• Finite field

• Polynomial ring

• Formal power series ring

Algebraic number theory

• Algebraic number field

• Integers modulo n

• Ring of integers

• p-adic integers

\mathbb {Z} _{p}

• p-adic numbers

\mathbb {Q} _{p}

• Prüfer p-ring

\mathbb {Z} (p^{\infty })

Noncommutative algebraNoncommutative rings

• Division ring

• Semiprimitive ring

• Simple ring

• Commutator

Noncommutative algebraic geometry

Free algebra

Clifford algebra

• Geometric algebra

Operator algebra

In algebra, a semiprimitive ring or Jacobson semisimple ring or J-semisimple ring is a ring whose Jacobson radical is zero. This is a type of ring more general than a semisimple ring, but where simple modules still provide enough information about the ring. Rings such as the ring of integers are semiprimitive, and an artinian semiprimitive ring is just a semisimple ring. Semiprimitive rings can be understood as subdirect products of primitive rings, which are described by the Jacobson density theorem.

Definition

A ring is called semiprimitive or Jacobson semisimple if its Jacobson radical is the zero ideal.

A ring is semiprimitive if and only if it has a faithful semisimple left module. The semiprimitive property is left-right symmetric, and so a ring is semiprimitive if and only if it has a faithful semisimple right module.

A ring is semiprimitive if and only if it is a subdirect product of left primitive rings.

A commutative ring is semiprimitive if and only if it is a subdirect product of fields, (Lam 1995, p. 137).

A left artinian ring is semiprimitive if and only if it is semisimple, (Lam 2001, p. 54). Such rings are sometimes called semisimple Artinian, (Kelarev 2002, p. 13).

Examples

The ring of integers is semiprimitive, but not semisimple.
Every primitive ring is semiprimitive.
The product of two fields is semiprimitive but not primitive.
Every von Neumann regular ring is semiprimitive.

Jacobson himself has defined a ring to be "semisimple" if and only if it is a subdirect product of simple rings, (Jacobson 1989, p. 203). However, this is a stricter notion, since the endomorphism ring of a countably infinite dimensional vector space is semiprimitive, but not a subdirect product of simple rings, (Lam 1995, p. 42).

References

Jacobson, Nathan (1989), Basic algebra II (2nd ed.), W. H. Freeman, ISBN 978-0-7167-1933-5
Lam, Tsit-Yuen (1995), Exercises in classical ring theory, Problem Books in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94317-6, MR 1323431
Lam, Tsit-Yuen (2001), A First Course in Noncommutative Rings, Berlin, New York: Springer-Verlag, ISBN 978-0-387-95325-0
Kelarev, Andrei V. (2002), Ring Constructions and Applications, World Scientific, ISBN 978-981-02-4745-4

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