principal ideal ring
In algebra, a 'principal ideal ring' is a type of ring where every ideal is generated by a single element, making it a fundamental concept in ring theory.
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Definition
C2Algebra
(technical, academic)A ring in which every ideal can be generated by a single element.
Example
- In a principal ideal ring, the structure of ideals is simpler to study because each ideal is generated by just one element.
C2Ring Theory
(technical, academic)A commutative ring where every ideal is of the form {ar : r in R} for some element a in R.
Example
- A commutative principal ideal ring allows for easier factorization of elements.
C2Ring Theory
(technical, academic)A noncommutative ring where every left and right ideal can be generated by a single element.
Example
- In noncommutative principal ideal rings, both left and right ideals are generated by one element, simplifying their analysis.
Similar
Terms that have similar or relatively close meanings to "principal ideal ring":
prime idealprime ringprimitive elementlocal ringprincipal partcommutative algebra