A ring with identity is said to be a local ring if it contains a unique maximal right ideal [2; 751. This implies that the unique maximal right ideal is also the unique maximal ideal. But rings with unique maximal ideals need not be local rings. The ring of all 2x2 matrices over the ring of integers modulo 4 has a unique maximal ideal but it is not a local ring since local rings do not contain proper idempotents. In this paper we shall exhibit many classes of rings which contain unique maximal ideals, investigate their interrelationship and find conditions when they are local rings. We assume that every ring under consideration is not a simple ring but contains identity. A right (two-sided) ideal in a ring is called large if it has non-zero intersection with every non-zero right (two-sided) ideal of the ring. A ring is called intersective if every non-zero ideal in it is large. We shall call a, ring to be the RE-, RL-or RI-ring if respectively every one of its proper homomorphic images is free from proper idempotents, a local ring or an intersective ring. A (right) valuation ring is a ring in which the (right) ideals form a chain. A ring is said to be bounded if every non-zero right ideal contains a nonzero two-sided ideal. For any right ideal P in a ring A, we denote P*={xEA: Px=O}. J ( A ) denotes the JACOBSON radical of the ring A .
We show that the class of RE-rings coincide with the class of RL-rings in commutative case. It is trivially true that RL-rings are always RE-rings. Moreover commutative RI-rings are RL-rings. But the example P[x, y, z]/(x2, y2, z2),
where F is a field, asserts that an RL-ring need not be RI-ring. In commutative case we prove that an RL-ring is a RI-ring if and only if every proper homomorphic image is a valuation ring. We discuss in a great length how these classes can be separated by determining their structures.