# Ring and Module Theory by Toma Albu, Gary F. Birkenmeier, Ali Erdogan, Adnan Tercan

By Toma Albu, Gary F. Birkenmeier, Ali Erdogan, Adnan Tercan

The quantity involves a suite of invited study papers and expository/survey articles, lots of that have been offered on the foreign convention on Ring and Module concept held at Hacettepe college in Ankara, Turkey, in August 2008. the chosen papers and articles study extensive ranging and leading edge advancements in a variety of parts of algebra, together with ring conception, module conception, and commutative algebra. there's additionally a bit dedicated to open difficulties to inspire additional examine.

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15]) The following are equivalent for a ring R. (i) R is semiprime and has exactly n minimal prime ideals. (ii) QIC (R) = RB(Q(R)) is a direct sum of n prime rings. (iii) QIC (R) = RB(Q(R)) ∼ = R/P1 ⊕ · · · ⊕ R/Pn , where each Pi is a minimal prime ideal of R. 2. 2. The independence of these deﬁnitions is beneﬁcial in the sense that they provide distinct tools for analyzing interconnections between a ring and its right essential overrings relative to a class K. Also the following example shows that there is a quasi-Baer ring R (hence R itself is a quasi-Baer right ring hull of R), but R does not have a unique right FI-extending right ring hull.

R. S. Roumanie 23 (71) (1979), 115–116. [3] T. Albu, On commutative Grothendieck categories having a Noetherian cogenerator , Arch. Math. (Basel) 34 (1980), 210–219. [4] T. Albu, Certain Artinian lattices are Noetherian. Applications to the relative Hopkins-Levitzki Theorem, in “Methods in Ring Theory”, edited by F. Van Oystaeyen, D. Reidel Publishing Company, Dordrecht (Holland), 1984, pp. 37–52. [5] T. Albu, Classes of lattices (co)generated by a lattice and their global (dual) Krull dimension, Discrete Math.

3, for any hereditary torsion theory τ = (T , F ) on Mod-R, and any module MR , the lattice Satτ (M ) is isomorphic to the lattice Sub(Tτ (M )) of all subobjects of the object Tτ (M ) in the quotient category Mod-R/T , where Tτ : Mod-R −→ Mod-R/T is the canonical functor. The same happens also in our latticial frame: if X is a Serre class for L ∈ M such that L has an X -closure, then L/ ∼ X Sat X (L). 22 T. Albu This implies that the lattice L is X -Noetherian (resp. X -Artinian) ⇐⇒ the lattice SatX (L) is Noetherian (resp.