| Abstract:
Let \( \mathbf{R} = (R;\, \cdot,\, +,\, -,\, 0,\, 1) \)
be a unitary commutative ring and
\(\mathbf{D} = (D; \wedge, \vee, 0, 1) \)
a bounded distributive lattice. Denote by
\( \mathrm{Id}\, \mathbf{R} \) (respectively,
\( \mathrm{Id}\, \mathbf{D} \))
the set of all ideals of \( \mathbf{R} \) (respectively, \( \mathbf{D} \)),
ordered by containment. Then
under this ordering,
\( \mathrm{Id}\, \mathbf{R} \)
(respectively, \( \mathrm{Id}\, \mathbf{D} \))
is a modular (respectively,
distributive) lattice. We consider each to be a multiplicative semilattice
\( \mathbf{M} = (M ; \vee, \cdot, 0, 1) \),
where \( \vee \) is the join of ideals, \( 0=\lbrace 0\rbrace \) is the
smallest ideal,
\( \mathbf{R} \) (respectively, \( \mathbf{D} \))
is the largest ideal, and \(\cdot \) is the
ideal product. For distributive lattice ideals \( I \), \( J \),
the product is just their intersection:
\( I \cdot J = I \cap J \). But for commutative ring ideals
\( I \), \( J \),
the product is the ideal
\( I \cdot J :=
\langle \lbrace i\cdot j \mid i \in I,\,j \in J \rbrace\rangle\).
Every multiplicative semilattice \(\mathbf{M} \) has a maximal quotient
\( \mathbf{M}/\Delta \)
which is a distributive lattice.
Let \( \mbox{SmPr}\, \mathbf{R} \) be the set of all semiprime
ideals of \(\mathbf{R}\), ordered by containment.
Theorem: Under ordering by containment,
\( \mbox{SmPr}\, \mathbf{R} \) is a distributive lattice,
and as multiplicative semilattices,
\( \mathbf{M}/\Delta \) is isomorphic to \( \mbox{SmPr}\, \mathbf{R} \).
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