In this paper \em commutative algebraic multiplication $m$-lattices \em are investigated. They are closely related to the Dedekind \em ideal structures \em of \em commutative rings with identity \em satisfying ${\mathfrak a}\supseteq {\mathfrak b}\Longrightarrow {\mathfrak a}\,\vert \,{\mathfrak b}$\,. Rings of this type were introduced as \em multiplication rings \em by {\sc Wolfgang Krull} and studied by {\sc Shinziro Mori} over a period of 25 years\,. Modifying notions of classical ideal theory, we succeed in carrying over classical ideal results to algebraic $m$-lattices satisfying for at least one \em generating system of compact elements \em the implication $ a \cdot U = a \Longrightarrow a \cdot U^* = 0\enspace(\exists\, U^* \,\bot\, U)\,.$ In particular we study the \em Pr\"ufer property \em $ a_1 + \ldots + a_n \supseteq B \Longrightarrow a_1 + \ldots + a_n \,\vert\, B\,$, \em the archimedean property \em $A^n\supseteq B\enspace(\forall\, n\in {\bf N})\Longrightarrow AB = B\,$, the \em kernel property $\ker A=A$, and the \em multiplication property, \em that is $A \supseteq B \Longrightarrow A\,\vert \, B\,.$