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Every ring that is semisimple as a module over itself has zero Jacobson radical, but not every ring with zero Jacobson radical is semisimple as a module over itself. A J-semisimple ring is semisimple if and only if it is an artinian ring, so semisimple rings are often called artinian semisimple rings to avoid confusion.
If R is a ring, then any R-module is an R-semimodule. Conversely, it follows from the second, fourth, and last axioms that (-1)m is an additive inverse of m for all , so any semimodule over a ring is in fact a module. Any semiring is a left and right semimodule over itself in the same way that a ring is a left and right module over itself.
This makes the analogy between ring and semiring on the one hand and group and semigroup on the other hand work more smoothly. These authors often use rig for the concept defined here. [ 1 ] [ a ] This originated as a joke, suggesting that rigs are ri n gs without n egative elements.
Semisimple rings are left and right hereditary via the equivalent definitions: all left and right ideals are summands of R, and hence are projective.By a similar token, in a von Neumann regular ring every finitely generated left and right ideal is a direct summand of R, and so von Neumann regular rings are left and right semihereditary.
The endomorphism ring of an Artinian module is a semilocal ring. Semi-local rings occur for example in algebraic geometry when a (commutative) ring R is localized with respect to the multiplicatively closed subset S = ∩ (R \ p i), where the p i are finitely many prime ideals.
Modern surface mount devices eliminate most of the drilled holes through circuit boards, and have short metal leads or pads on the package that can be secured by oven-reflow soldering. Aerospace devices in flat packs may use flat metal leads secured to a circuit board by spot welding , though this type of construction is now uncommon.
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