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Principal Ring


For some authors (e.g., Bourbaki, 1964), the same as principal ideal domain. Most authors, however, do not require the ring to be an integral domain, and define a principal ring (sometimes also called a principal ideal ring) simply as a commutative unit ring (different from the zero ring) in which every ideal is principal, i.e., can be generated by a single element. Examples include the ring of integers Z, any field, and any polynomial ring in one variable over a field. While all Euclidean rings are principal rings, the converse is not true.

If the ideal I of the commutative unit ring R is generated by the element a of R, in any quotient ring R^_ the corresponding ideal I^_ is generated by the residue class a^_ of a. Hence, every quotient ring of a principal ideal ring is a principal ideal ring as well. Since Z is a principal ideal domain, it follows that the rings Z_n are all principal ideal rings, though not all of them are principal ideal domains.

Principal ideal rings which are not domains have abnormal divisibility properties. For example, in Z_6, the identities

 2^_·5^_=4^_,    4^_·5^_=2^_

and

 2^_·2^_=4^_,    4^_·2^_=2^_

show that two elements (2^_,4^_) which divide each other can differ both by an invertible (5^_) and a noninvertible factor (2^_). Moreover, a prime element need not be irreducible. For example, if 2^_ divides the product of two factors of Z_6, one of these is certainly the residue class of an even number, i.e., it is a multiple of 2^_. Hence 2^_ is prime. On the other hand, in the decomposition 2^_=2^_·4^_, none of the factors is invertible, which shows that 2^_ is not irreducible.

For such reasons, many authors refrain from extending the divisibility notion and the related concepts from principal ideal domains to principal ideal rings.

Principal rings are very useful because in a principal ring, any two nonzero elements have a well-defined greatest common divisor. Furthermore each nonzero, nonunit element in a principal ring has a unique factorization into prime elements (up to unit elements).


See also

Euclidean Ring, Irreducible Element, Prime Element, Principal Ideal

Portions of this entry contributed by Margherita Barile

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References

Bourbaki, N. "Anneaux Principaux." §7.1 in Eléments de Mathématiques, Livre II: Algèbre, 2ème ed. Paris, France: Hermann, 1964.Wilson, J. C. "A Principal Ring that is Not a Euclidean Ring." Math. Mag. 34-38, 1973.

Referenced on Wolfram|Alpha

Principal Ring

Cite this as:

Barile, Margherita and Weisstein, Eric W. "Principal Ring." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PrincipalRing.html

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