This is an example of a finiteness condition which, at least historically, has dominated in ring theory.

## Serial Rings

Ifwe would like to consider a requirement of a lattice-theoretical type, other than being Artinian or Noetherian, the most natural is uni-seriality. Here a module M is called uni-serial if its lattice of submodules is a chain, and a ring R is uni-serial if both RR and RR are uni-serial modules.

The class of uni-serial rings includes commutative valuation rings and closed under homomorphic images. But it is not closed under direct sums nor with respect to Morita equivalence: a matrix ring over a uni-serial ring is not uni-serial. There is a class of rings which is very close to uni-serial but closed under the constructions just mentioned: serial rings. Amongst others this class includes triangular matrix rings over a skew field.

Also if F is a finite field of characteristic p and G is a finite group with a cyclic normal p-Sylow subgroup, then the group ring FG is serial. Serial Prime Goldie Rings. Therefore, ideals ePie and ePje of the ring eAe are comparable with respect to inclusion. Since condition 3 is symmetrical, it is sufficient to prove 2 3. Since A is a right Noetherian right semihereditary ring, A is a right hereditary ring. The semiperfect prime right hereditary ring A is a right serial ring by L e m m a 1.

Since A is a prime ring, its nonzero ideal B is an essential left ideal. Then A l P is a left Artinian ring; this is a contradiction. The left Noetherian left semihereditary ring A is left hereditary. Then the following assertions hold: 1 if M is a right ideal in A such that MA is a distributive resp. Assume t h a t MA is a distributive module. By Lemma 2.

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Assume that MA is a uniform module. Then the module MQQ is uniform. Therefore, MQQ is a uniserial module. Assertions 2 and 3 follow from 1. Let the identity element of a ring A be a sum of nonzero idempotents el,. Therefore, there exists a nonzero element ejc E ejA such that 0 ejyej ejc e ejbA. Assume the contrary. Then either eivej E Lij or elvej E Rij. Assume that eivej E L i j. Similarly, the inclusion eivej E Rij is impossible. Therefore, eivej E Tij.

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The inclusion ejwei E Tjl can be proved in a similar way. Then M is a Noetherian resp. Artinian module. Then A is a right Artinian ring. Artinian modules is a Noetherian resp. Artinian module, A is a either right Noetherian right perfect ring or a right Artinian ring by 1. Note that any right Noetherian right perfect ring is right Artinian. Hence R is a right Artinian ring.

By 1 , all ejAei are Artinian right eiAel-modules. Therefore, all ejAei are Artinian right R-modules. Hence A is an Artinian right module over its unitary subring R. Therefore, A is a right Artinian ring. There exists an rn such that 7rm is the identity p e r m u t a t i o n. Then M u a v m is a diagonal matrix. Let A be a uniseriaI ring with the maximum condition on right annihilators. Then the following assertions hold: 1 A is a ring with the maximum condition on left annihilators; 2 A has the classical ring of quotients Q which is a uniserial Artinian ring.

Since A is a ring with the m a x i m u m condition on right annihilators, the right singular ideal N is nilpotent by L e m m a 1. W i t h o u t loss of generality, we may assume t h a t N is a nonzero ideal. Therefore, D properly contains N. In addition, A is a uniserial ring. Then A has the classical ring of quotients Q which is a uniserial ring by L e m m a 2. Therefore, Q is a uniserial semiprimary ring. Then Q is an Artinian ring.

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Then the following assertions hold" 1 T is a Ore set in A , the ring of quotients Q of the ring A with respect to T is a serial ring, all eiO. Let us prove that for any i and j , there exists an element eju i, j ej E Tjj such that eiaej eju i, j e j E eiteiA. Therefore, ejbei eiaej E Tjj. Therefore, T is a right Ore set. Similarly, T is a left Ore set. Therefore, the ring of quotients Q of A with respect to T exists. The remaining assertions are directly verified.

By 1 , T is a right Ore set. Therefore, R is a right Ore set. Similarly, R is a left Ore set. Therefore, the classical ring of quotients Qcl A of the ring A exists. The serial ring of quotients Q is contained in Q d A. Let us prove the inclusion Qd A C Q. It is sufficient to prove that any regular element q of the ring Q is invertible in Q.

Since Q is a serial ring, L e m m a 2. All the elements ei zqy ei are regular in eiQei. Therefore, zqy k is invertible in Q. In addition, z and y are invertible in Q. Then q is invertible in Q. The remaining assertions follow from 2. Theorem 2 follows from L e m m a 2. For a finitely generated projective right module Q over a serial ring A , the following assertions hold.

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The module P is isomorphic to a direct sum of local modules over the uniserial ring A. Therefore, P is serial. Then h S is a singular module. Every finitely generated module over a serial ring is the sum of a singular finitely generated module and a serial projective finitely generated module the sum is not necessarily a direct surrg. The proposition follows from L e m m a 3.

Let A be a serial ring. Then the following assertions hold: 1 every finitely generated nonsingular A-module is a serial projective module [5]; 2 every indecomposable finitely generated nonsingular A-module is a uniserial projective cyclic mod- ule; 3 every uniform nonsingular A-module M is a uniserial module, and consequently, all indecompos- able injective nonsingular A-modules are uniserial modules [13].

Assertion 1 follows from Proposition 3. Assume that M is a right module over a ring A , and M is not a singular module. Then M contains an isomorphic copy of some nonzero right ideal of the ring A. In addition, if A is a right finite-dimensional ring, then M contains an isomorphic copy of a nonzero uniform right ideal of A. By assumption, there exists a nonzero element m E M such that the right ideal r m of the ring A is not essential.

If A is right finite-dimensional, then B contains a nonzero uniform right ideal C , whence M contains h C. Let A be a right finite-dimensional ring. Since A is right finite-dimensional, B A is an essential extension of a finitely generated right ideal C of the ring A. Since N is a finite direct sum of injective modules, N is injective. In addition, f - g B is a submodule of the nonsingular module M.

Then g is an extension of f , whence the module M is injective. Therefore, any nonzero direct summand of the injective module M has a nonzero uniform injective direct summand. By 1 , the module N is injective. The first assertion follows from Proposition 3. The fact that every nonsingular injective right module over a serial ring is a serial module follows from Lemma 3. Let N be a arbitrary nonsingular A-module. By Corollary 3.

## Serial Rings

Then N is a flat module see [14, References I. Singh, "Serial right Noetherian rings," Canad. Tuganbaev, "Distributive rings and modules," Mat. Zametld [Math. Notes], 47, No. Warfield, "Bezout rings and serial rings," Comm.

## Serial Rings, Book by G. Puninski (Paperback) | obeqiciqypit.tk

Algebra, 7, Warfield, "Serial rings and finitely presented modules," J. Algebra, 37, No. Goodearl, P,. Faith, Algebra, Vol. Small, "Semihereditary rings," Buff. Chatters, "The restricted minimum condition in Noetherian hereditary rings," J. London Math. Tuganbaev, " Right or left distributive rings," Mat. Zametkl [Math.