Simple element

A simple element - a generalization of the concept of a prime number to the case of an arbitrary commutative monoid with two-sided cancellation , is defined as a nonzero element nonzero element $p\in G$ ${\ displaystyle p \ in G}$ $p \ in G$ such that the product $a b$ ${\ displaystyle ab}$ $ab$ can be divided into $p$ ${\ displaystyle p}$ $p$ only when at least one of the elements $a$ ${\ displaystyle a}$ $a$ or $b$ ${\ displaystyle b}$ $b$ divided by $p$ ${\ displaystyle p}$ $p$ .

A simple element is always irreducible ; in the general case, simplicity does not follow from irreducibility, but in a Gaussian semigroup the concepts of irreducibility and simplicity coincide, and moreover, if every irreducible element from $G$ ${\ displaystyle G}$ $G$ is simple then the semigroup $G$ ${\ displaystyle G}$ $G$ - Gaussian.

The concept is naturally transferred to the domain of integrity , in this case there is an equivalence of the irreducibility and simplicity of the element for factorial (Gaussian) rings , and from the simplicity of all irreducible elements in the domain of integrity it follows that the ring is factorial. In addition, the simplicity of an element is equivalent to the simplicity of the main ideal generated by it.

There are also generalizations of the concepts of simplicity and irreducibility to the non-commutative case.

Literature

Cohn P. Free rings and their connections. - M. , 1975.
Kurosh A.G. Lectures on general algebra. - 2nd ed. - M .: Fizmatlit, 1973.
A simple element is an article from the Mathematical Encyclopedia . O. A. Ivanova
Shevrin L.N. Chapter IV. Semigroups // General Algebra / Subtotal. ed. L.A. Skornyakova . - M .: Nauka , 1991. - T. 2. - S. 11—191. - 480 p. - (Reference Mathematical Library). - 25,000 copies. - ISBN 5-9221-0400-4 .

Simple element

Literature

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