Primitive ring

In the branch of abstract algebra known as ring theory, a left primitive ring is a ring which has a faithful simple left module. Well known examples include endomorphism rings of vector spaces and Weyl algebras over fields of characteristic zero.

In the branch of abstract algebra known as ring theory, a left primitive ring is a ring which has a faithful simple left module. Well known examples include endomorphism rings of vector spaces and Weyl algebras over fields of characteristic zero. Another equivalent definition states that a ring is left primitive if and only if it is a prime ring with a faithful left module of finite length ( , Ex. 11.19, p. 191). ==Properties== One-sided primitive rings are both semiprimitive rings and prime rings. Since the product ring of two or more nonzero rings is not prime, it is clear that the product of primitive rings is never primitive. For a left Artinian ring, it is known that the conditions "left primitive", "right primitive", "prime", and "simple" are all equivalent, and in this case it is a semisimple ring isomorphic to a square matrix ring over a division ring. More generally, in any ring with a minimal one sided ideal, "left primitive" = "right primitive" = "prime". A commutative ring is left primitive if and only if it is a field. Being left primitive is a Morita invariant property. ===Full linear rings=== A special case of primitive rings is that of full linear rings. A left full linear ring is the ring of all linear transformations of an infinite-dimensional left vector space over a division ring. (A right full linear ring differs by using a right vector space instead.) In symbols, R=\mathrm{End}(_D V) where V is a vector space over a division ring D. It is known that R is a left full linear ring if and only if R is von Neumann regular, left self-injective with socle soc( R R) ≠ {0}. Through linear algebra arguments, it can be shown that \mathrm{End}(_D V)\, is isomorphic to the ring of row finite matrices \mathbb{RFM}_I(D)\, , where I is an index set whose size is the dimension of V over D. Likewise right full linear rings can be realized as column finite matrices over D. Using this we can see that there are non-simple left primitive rings. By the Jacobson Density characterization, a left full linear ring R is always left primitive. When dim D V is finite R is a square matrix ring over D, but when dim D V is infinite, the set of finite rank linear transformations is a proper two-sided ideal of R, and hence R is not simple.

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