Abstract
We prove that if R is a right semi-Artinian ring, then R is an exchange ring and every irredundant set Simp R of representatives of simple right R -modules carries a canonical structure of an Artinian poset, which is a Morita invariant. We investigate several basic features of this order structure and, for a wide class of right semi-Artinian rings, which we call nice , we establish a link between those (two-sided) ideals which are pure as left ideals and some upper subsets of Simp R . If R is nice and Simp R does not contain infinite antichains, then that link realizes an anti-isomorphism from the lattice of upper subsets of Simp R to the set of all ideals which are pure as left ideals. Further we show that every Artinian poset (possibly after adding a suitable maximal element if it is infinite) is order isomorphic to Simp R for some nice right semi-Artinian ring R .
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