  Let $R$ be a commutative ring with identity. The structure theorem says that
$R$ is a PIR (resp., UFR, general ZPI-ring, $\pi$-ring) if and only if $R$ is a
finite direct product of PIDs (resp., UFDs, Dedekind domains, $\pi$-domains)
and special primary rings. All of these four types of integral domains are
Krull domains, so motivated by the structure theorem, we study the prime
factorization of ideals in a ring that is a finite direct product of Krull
domains and special primary rings. Such a ring will be called a general Krull
ring. It is known that Krull domains can be characterized by the star
operations $v$ or $t$ as follows: An integral domain $R$ is a Krull domain if
and only if every nonzero proper principal ideal of $R$ can be written as a
finite $v$- or $t$-product of prime ideals. However, this is not true for
general Krull rings. In this paper, we introduce a new star operation $u$ on
$R$, so that $R$ is a general Krull ring if and only if every proper principal
ideal of $R$ can be written as a finite $u$-product of prime ideals. We also
study several ring-theoretic properties of general Krull rings including
Kaplansky-type theorem, Mori-Nagata theorem, Nagata rings, and Noetherian
property.
