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Among theorems relating to a PID, The main a single is the construction theorem for finitely generated modules in excess of a principal great area.

of a finite group G over a field k is semisimple if the characteristic of k will not divide the purchase of G. (Maschke's theorem)

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may be the unique ring homomorphism. Equally as from the team case, just about every ring could be represented to be a quotient of the absolutely free ring.[40]

An algebra more than a subject k is artinian if and only if it's finite dimension. Consequently, a semisimple algebra more than a subject is necessarily finite-dimensional, even though a simple algebra can have infinite dimension; e.g., the ring of differential operators.

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. One example of a nilpotent component is actually a nilpotent matrix. A nilpotent factor in a very nonzero ring is essentially a zero divisor.

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The valuation ring of v is the subring of K consisting of zero and all nonzero f this kind of that v(f) ≥ 0.

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