Exchange property and the natural preorder between simple modules over semi-Artinian rings

We prove that if R is a right semi-Artinian ring, then R is an exchange ring and every irredundant set SimpR 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 SimpR. If R is nice and SimpR does not contain infinite antichains, then that link realizes an anti-isomorphism from the lattice of upper subsets of SimpR 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 SimpR for some nice right semi-Artinian ring R.