Full Linear Rings Research Articles

On products of full linear rings

On products of full linear rings

Rings Without a Middle Class from a Lattice-Theoretic Perspective

We further the study of rings with no middle class by focusing on an interpretation of that property in terms of the lattice of hereditary pretorsion classes over a given ring. For non-semisimple rings, the absence of a middle class is equivalent to the requirement that the class of all semisimple right modules be a coatom in that lattice. Taking advantage of this perspective, we discover new facts and shed light on others already known with a possibly more direct interpretation without having to refer to an exhaustive analysis of the structure theorems available in the literature. Our approach also allows us to characterize rings with no middle class in terms of hereditary pretorsion classes containing the class of all singular right modules. We discuss the open problem of whether there is a ring with no right middle class which is not right Noetherian and see, in particular, that an indecomposable ring satisfying that property would have to be Morita equivalent to a certain type of subring of a full linear ring.

A new setting for constructing von Neumann regular rings

ABSTRACTA classical and clever construction by George Bergman in the mid 1970s was of a unit-regular algebra Q (over an arbitrary field F) that contains a regular subalgebra R which is not unit-regular. The algebra Q was constructed inside a full linear ring of an uncountable-dimensional vector space over F. Here we give a much simpler construction of Bergman’s R and Q inside an algebra B of countably-infinite matrices. The algebra B does not seem to have been noticed before, possibly because it is not obviously a ring. It also suggests possibilities for answering the difficult open problem of constructing a non-separative regular ring.

Semiartinian V-Rings and Semiartinian Von Neumann Regular Rings

Semiartinian right V-rings, which we call right SV-rings, form a special class of Von Neumann regular rings. We characterize these rings by the fact that every factor ring imbeds as a subring in a direct product of right full linear rings containing the socle. If R is a semiartinian ring with all primitive factor rings artinian, then the condition of being an SV-ring is right/left symmetrical and is equivalent to being regular. On the other hand, if R is a right and left SV-ring, then all primitive factor rings of R are artinian. For right SV-rings whose proper ideals are prime we show that the condition of being unit-regular is equivalent to being directly finite. On the other hand we show that there exists a directly finite right SV-ring which is not unit-regular. Furthermore we provide two constructions. For any given ordinal ξ, the first one gives a prime, unit-regular right SV-ring of Loewy length ξ + 1, which is not a left V-ring, and is hereditary if ξ is a natural number; the second one gives a directly infinite right SV-ring, not a left V-ring, whose Loewy length is ξ + 2 if ξ is a natural number and is ξ + 1 otherwise. These constructions are general enough to produce a wide supply of SV-rings, starting from given ones.

Structure of pseudo-semisimple rings

AbstractA ring R is called right pseudo-semisimple if every right ideal not isomorphic to R is semisimpie. Rings of this type in which the right socle S splits off additively were characterized; such a ring has S2 = 0. The existence of right pseudo-semisimple rings with zero right singular ideal Z remained open, except for the trivial examples of semisimple rings and principal right ideal domains. In this work we give a complete characterization of right pseudo-semisimple rings with S2 = 0. We also give examples of non-trivial right pseudo-semisimple rings with Z = 0; in fact it is shown that such rings exist as subrings in every infinite-dimensional full linear ring. A structure theorem for non-singular right pseudo-semisimple rings, with homogeneous maximal socle, is given. The general case is still open.

The structure of rings with faithful nonsingular modules

It is shown that the existence of a faithful nonsingular uniform module characterizes rings which have a full linear maximal quotient ring. New information about the structure of these rings is obtained and their maximal quotient rings are constructed in an explicit manner. More generally, rings whose maximal quotient rings are finite direct sums of full linear rings are characterized by the existence of a faithful nonsingular finite dimensional module.

Maximal quotient rings of prime group algebras

It is shown that a prime nonsingular group algebra KG, where G is a group whose conjugacy classes are countable, has for its maximal right quotient ring either a full linear ring or a simple directly infinite ring. In the case where G is also locally finite only the second type can occur.

Right Orders in Full Linear Rings

Right Orders in Full Linear Rings

Right orders in full linear rings

In this paper a right orderRRin an infinite dimensional full linear ring is characterized as a ring satisfying: (1)RRis meet-irreducible (with zero right singular ideal) and contains uniform right ideals; (2) the closed right ideals ofRRare right annihilator ideals, and each such right ideal is essentially finitely generated; (3)RRpossesses a reducing pair (i.e. a pair(β1,β2)({\beta _1},{\beta _2})of elements for whichβ1R,β2R{\beta _1}R,{\beta _2}Randβ1r+β2r\beta _1^r + \beta _2^rare large right ideals ofRR); (4) for eacha∈Ra \in Rwithal=0,aR{a^l} = 0,aRcontains a regular element ofRR. A second characterization ofRRis also given. This is in terms of the right annihilator ideals ofRRwhich have the same (uniform) dimension asRR{R_R}. The problem of characterizing right orders in (infinite dimensional) full linear rings was posed by Carl Faith. The Goldie theorems settled the finite dimensional case.

On some rings whose modules have maximal submodules

It is shown that a principal right ideal domain, having the property that every right module has a maximal submodule must be simple. Strong conditions satisfied by these rings are deduced giving evidence for the conjecture that they must be V-rings. We also generalize an example of Faith by showing that a subring of an infinite dimensional full linear ring, which contains the socle of that is never a left V-ring. Cozzens [1] and Kolfmann [31 gave striking examples of simple principal right ideal domains which are right V-rings (every simple module is injective). These examples answer two questions of Faith, whether there exists a nonsemisimple, simple, noetherian V-ring, and whether every V-ring is regular; and Bass' question, whether a over which every right module has a maximal submodule (a max ring) must be right perfect. Interest in these examples is enhanced by the fact that, being simple domains, they are highly nonregular and highly nonperfect, thus answering the last two questions, very reasonable ones given the state of knowledge at the time, very definitively in the negative. They show that in the noncommutative case (see [31 for the commutative case) the hypothesis R is a max ring does not seem to place very strong conditions on R. Now the most general proof of the fact that these rings are simple comes from a theorem of Faith [2, p. 1301, that any V-ring which is an order in a semisimple must be a product of simple rings. Our first purpose is to show that in fact if is a right P.I.D., as the examples cited are, then one needs only assume that is a right max in order to conclude that is simple. This result seems interesting because it seems to be an exception to the apparent weakness of the max hypothesis as cited in the previous paragraph. Received by the editors April 19, 1974. AMS (MOS) subject classifications (1970). Primary 16A04, 16A30; Secondary 16A42, 16A46.

Prime regular rings of quotients

Let R be a ring with 1, and Q its maximal ring of right quotients. We give a number of necessary and sufficient conditions on R so that q is prime regular. We obtain as corollaries conditions so that Q is a full linear ring or Q is simple and satisfies xy = 1 implies yx = 1.

Primeness of right orders in full linear rings

Primeness of right orders in full linear rings

Subdirect sum decompositions of endomorphism rings

In [4] Levy defined irredundant subdirect sums of rings. This provided a unique decomposition of a left quotient semisimple ring as a subdirect sum of left quotient simple rings. He also showed that irredundant subdirect sums of prime rings possess some rather strong uniqueness properties. In [2] essential subdirect sums were defined and Levy's decomposition theorem was obtained for essential subdirect sums. Furthermore in [3] it was shown that a ring R has a maximal left quotient ring which is a direct product of full linear rings if and only if R is an essential subdirect sum of rings whose maximal left quotient ring is full linear. In §1 of this paper we show that for subdirect sums of prime rings, essential and irredundant subdirect sums are identical. We also give some uniqueness properties of essential subdirect sums which are even stronger than those for irredundant subdirect sums. In §2 we show that if M is a torsionless 12-module and 22 is an essential subdirect sum of prime rings, then the endomorphism ring of M is also an essential subdirect sum of prime rings. We get this theorem as a consequence of a more general result about Morita contexts. The proof of the more general result is notationally more efficient. 1* Essential subdirect sums of prime rings* In what follows R will be an associative ring which is not assumed to have an identity. RM will express that M is a left JS-module. A submodule K of M is an essential submodule if every nonzero submodule of M has nonzero intersection with K. M will then be an essential extension

Right orders in full linear rings

Right orders in full linear rings

Quotient Full Linear Rings

We define a ring $R$ to be an FL (full linear) ring if $R$ is isomorphic to the full ring of linear transformations of a left vector space over a division ring. $R$ is QFL if its left maximal quotient ring is an FL ring. In this paper we give necessary and sufficient conditions for a ring to be a QFL ring. We also generalize some results of Chase and Faith concerning subdirect sum decompositions of rings whose left maximal quotient ring is the direct product of FL rings.

Quotient full linear rings

We define a ring R R to be an FL (full linear) ring if R R is isomorphic to the full ring of linear transformations of a left vector space over a division ring. R R is QFL if its left maximal quotient ring is an FL ring. In this paper we give necessary and sufficient conditions for a ring to be a QFL ring. We also generalize some results of Chase and Faith concerning subdirect sum decompositions of rings whose left maximal quotient ring is the direct product of FL rings.

Cyclic Injective Modules of Full Linear Rings

Cyclic Injective Modules of Full Linear Rings

Cyclic injective modules of full linear rings

Cyclic injective modules of full linear rings

Quotient rings and direct products of full linear rings

Quotient rings and direct products of full linear rings