Subring

Abstract: One finds in the literature a thorough-going discussion of rings without radical and with minimal condition for left ideals (semi-simple rings). For the structure of a ring whose quotient-ring with respect to the radical is semi-simple one can refer to the investigations of K6the (see K below). In this paper we shall examine the structure of a ring A with radical R D 0 and with minimal condition for left ideals ("general" MLI ring). The key-stone of our investigations is the fact that the radical of A is nilpotent, and this result we shall establish in ?1. In ?2 we shall prove that the sum of all minimal non-zero left ideals is a completely-reducible left ideal 9A, and in ?3 we shall examine the distribution of idempotent and nilpotent left ideals in 9)1. In ??4-6 we shall discuss the two "extreme cases": (1) when A is nilpotent, and (2) when A is idempotent. For a non-nilpotent A we shall prove that the existence of either a right-hand or a left-hand identity is sufficient for the existence of a composition series of left ideals of A. If A is any MLI ring, one can find a smallest exponent k such that Ak = Ak+l = ... . In ?7 we show that A is the sum of Ak (which is idempotent) and a nilpotent MLI ring. We wish to emphasize the fact that A is to be regarded throughout as a ring without operators. In ?8, however, we shall see that some of our most interesting results are valid for operator domains of a certain type. We conclude the Introduction with an explanation of our notation and terminology. Rings and subrings will usually be denoted by roman capitals; we shall use gothic letters when it is desirable to emphasize the fact that a subring is an ideal. By the statement "a is a left (right) ideal of A" we shall mean that a is an additive abelian group which admits the elements of A as left-hand (right-hand) operators. Observe that our definition does not' imply that a is a subring of A. The term "left ideal," with no qualifying phrase, will always mean "left ideal of the basic ring." A ring with minimal condition for left (right) ideals which are contained in itself will be called an MLI (MRI) ring. Finally we point out that if a and b are subrings of A, then [a, b] denotes the cross-cut of a and b, while (a, b) represents the compound (join) of a and b-i.e.

Abstract: Given a ring $A$ and an $A$-coring $\cC$ we study when the forgetful functor from the category of right $\cC$-comodules to the category of right $A$-modules and its right adjoint $-\otimes_A\cC$ are separable. We then proceed to study when the induction functor $-\otimes_A\cC$ is also the left adjoint of the forgetful functor. This question is closely related to the problem when $A\to {}_A{\rm Hom}(\cC,A)$ is a Frobenius extension. We introduce the notion of a Galois coring and analyse when the tensor functor over the subring of $A$ fixed under the coaction of $\cC$ is an equivalence. We also comment on possible dualisation of the notion of a coring.