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The following equivalent definitions of a left perfect ring R are found in Anderson and Fuller: [2]. Every left R-module has a projective cover.; R/J(R) is semisimple and J(R) is left T-nilpotent (that is, for every infinite sequence of elements of J(R) there is an n such that the product of first n terms are zero), where J(R) is the Jacobson radical of R.
Let C be an additive category, or more generally an additive R-linear category for a commutative ring R. We call C a Krull–Schmidt category provided that every object decomposes into a finite direct sum of objects having local endomorphism rings. Equivalently, C has split idempotents and the endomorphism ring of every object is semiperfect.
A ring is directly irreducible if and only if 0 and 1 are the only central idempotents. A ring R can be written as e 1 R ⊕ e 2 R ⊕ ... ⊕ e n R with each e i a local idempotent if and only if R is a semiperfect ring. A ring is called an SBI ring or Lift/rad ring if all idempotents of R lift modulo the Jacobson radical.
A prong set diamond in a gold ring.. Prong setting or prong mount refers to the use of metal projections or tines, called "prongs", to secure a gemstone to a piece of jewelry.A prong setting is one component of what is known to jewelers as a head, a claw-shaped type of binding (typically three, four, or six individual prongs per head) that is welded or soldered to a jewelry item to mount (or ...
A minimalist wedding ring is refined and intentional, offering an effortless sophistication. Thin bands, bezel settings, and subtle shining accents create a modern yet meaningful look. Key features:
When many stones are set closely together in this fashion, roughly 1 millimetre (0.039 in) apart, [7] covering a surface, this is known as a "pavé" setting, from the French for "paved" or "cobblestoned". When a long line is engraved into the metal going up to each of the beads, this is known as a "star setting".
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The ring R is called a semiprime ring if the zero ideal is a semiprime ideal. In the commutative case, this is equivalent to R being a reduced ring, since R has no nonzero nilpotent elements. In the noncommutative case, the ring merely has no nonzero nilpotent right ideals. So while a reduced ring is always semiprime, the converse is not true. [1]