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 
, 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 
of the commutative unit ring 
is generated by the element 
of 
, then in any nonzero quotient ring 
the corresponding ideal 
is generated by the residue class 
of 
. Hence, every nonzero quotient ring of a principal ideal ring
is a principal ideal ring as well. Since 
is a principal ideal domain, it follows that the rings 
for 
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 
,
the identities
 |
and
 |
show that two elements 
which divide each other can differ both by an invertible (
) and a noninvertible factor (
). Moreover, a prime element
need not be irreducible. For example, if 
divides the product of two factors
of 
,
one of these is certainly the residue class of an even number, i.e., it is a multiple
of 
.
Hence 
is prime. On the other hand, in the decomposition 
, none of the factors is invertible, which shows
that 
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. This does not imply unique
factorization of elements when zero
divisors are allowed. For example, in 
the class 
is a prime element, but
, so even the
number of prime factors is not uniquely determined. The unique
factorization domain theorem applies to principal
ideal domains, where zero divisors are excluded
(Lorscheid 2020, p. 32).
