ISRAEL JOURNAL OF MATHEMATICS 216 (2016), 415–440 DOI: 10.1007/s11856-016-1415-5 RINGS WITH LINEARLY ORDERED RIGHT ANNIHILATORS BY Greg Marks Department of Mathematics and Computer Science, St. Louis University, St. Louis, MO 63103, USA e-mail: marks@member.ams.org AND Ryszard Mazurek Faculty of Computer Science, Bialystok University of Technology, Wiejska 45A, 15–351 Bia lystok, Poland e-mail: r.mazurek@pb.edu.pl ABSTRACT We introduce the class of lineal rings, defined by the property that the lattice of right annihilators is linearly ordered. We obtain results on the structure of these rings, their ideals, and important radicals; for instance, we show that the lower and upper nilradicals of these rings coincide. We also obtain an affirmative answer to the K¨ othe Conjecture for this class of rings. We study the relationships between lineal rings, distributive rings, B´ ezout rings, strongly prime rings, and Armendariz rings. In particular, we show that lineal rings need not be Armendariz, but they fall not far short. 1. Introduction A good deal of the structure of a ring can often be determined from the lattice structure of its right (or left) ideals. Some preeminent cases include the theories of noetherian rings, von Neumann regular rings, local rings, Goldie dimension, 2-firs, uniserial rings, and rings whose right ideals form a distributive lattice. Received June 9, 2015 and in revised form December 4, 2015 415