Proper class generated by submodules that have supplements

Abstract

Bu tezde, $/Im /alpha/,$' n{/i}n $B$'de bir t/`{u}mleyeni, yani $/{V/subseteq BV+/Im/alpha=B/}$ k/`{u}mesinin minimum eleman{/i} bulunacak /c{s}ekilde t/`{u}m $/ensuremath{/xymatrix{0 & A /ar[r]^{/alpha} & B /ar[r] & C /ar[r] & 0}}$ k{/i}sa tam dizilerinin$/ensuremath/mathcal{S}$ s{/i}n{/i}f{/i}n{/i} in/-ce/-li/-yo/-ruz. $/Ext_{R}(C,A)$' n{/i}n bu dizilere kar/c{s}{/i}l{/i}k gelen elemanlar{/i}na $/kappa$-elemanlar denir. Genelde $/kappa$-elemanlar bir /`{o}z s{/i}n{/i}f olu/c{s}turmayabilir, fakat $R$ Dedekind b/`{o}lgesi /`{u}zerindeki burulma mod/`{u}llerinin $/mathcal{T}_{R}$ kategorisinde $/ensuremath/mathcal{S}$ bir /`{o}z s{/i}n{/i}ft{/i}r; s{/i}f{/i}rdan farkl{/i} $/ensuremath/mathcal{S}$-projektif mod/`{u}ller bulunmaz, $/ensuremath/mathcal{S}$-injektif mod/`{u}ller sadece injektif mod/`{u}llerdir. Tezde $/mathcal{T}_{R}$ kategorisinde $/ensuremath/mathcal{S}$-e/c{s}injektif mod/`{u}llerin yap{/i}s{/i}n{/i} da verdik. Ayr{/i}ca $/Im /alpha$'n{/i}n $B$'de $V$ diye bir t/`{u}mleyeninin bulundu/v{g}u ve $V/cap /Im /alpha/,$' n{/i}n s{/i}n{/i}rl{/i} oldu/v{g}u $/ensuremath{/xymatrix{0 & A /ar[r]^{/alpha} & B /ar[r] & C /ar[r] & 0}}$ k{/i}sa tam dizilerinin $/ensuremath{/mathcal{SB}}$ s{/i}n{/i}f{/i}n{/i} tan{/i}mlad{/i}k. $/Ext_{R}(C,A)$' n{/i}n bu dizilere kar/c{s}{/i}l{/i}k gelen elemanlar{/i}na $/beta$-elemanlar denir. Krull boyutu 1 olan Noether taml{/i}k b/`{o}lgesi /`{u}zerinde $/ensuremath{/mathcal{SB}}$' nin bir /`{o}z s{/i}n{/i}f olu/c{s}turdu/v{g}unu g/`{o}sterdik. $R$ Dedekind b/`{o}lgesi /`{u}zerinde burulma mod/`{u}llerinin $/mathcal{T}_{R}$ kategorisinde $/ensuremath{/mathcal{SB}}$ bir /`{o}z s{/i}n{/i}ft{/i}r; s{/i}f{/i}rdan farkl{/i} $/ensuremath{/mathcal{SB}}$-projektif mod/`{u}ller bulunmaz, $/ensuremath{/mathcal{SB}}$-injektif mod/`{u}ller sadece injektif mod/`{u}llerdir. $/mathcal{T}_{R}$kategorisinde indirgenmi/c{s} $/ensuremath{/mathcal{SB}}$-e/c{s}injektif mod/`{u}ller tam olarak s{/i}n{/i}rl{/i} mod/`{u}llerdir.

In this thesis, we study the class $/ensuremath/mathcal{S}$ of all short exact sequences $/ensuremath{/xymatrix{0 & A /ar[r]^{/alpha} & B /ar[r] & C /ar[r] & 0}}$ where $/Im /alpha$ has a supplement in $B$, i.e. a minimal element in the set $/{V/subseteq B/mid V + /Im /alpha = B/}$. The corresponding elements of $/Ext_{R}(C,A)$ are called $/kappa$-elements. In general $/kappa$-elements need not form a subgroup in $/Ext_{R}(C,A)$, but in the category $/mathcal{T}_{R}$ of torsion $R$-modules over aDedekind domain $R$, $/ensuremath/mathcal{S}$ is a proper class; there are no nonzero $/ensuremath/mathcal{S}$-projective modules and the only $/ensuremath/mathcal{S}$-injective modules are injective $R$-modules in $/mathcal{T}_{R}$. In this thesis we also give the structure of $/ensuremath/mathcal{S}$-coinjective $R$-modules in $/mathcal{T}_{R}$. Moreover, we define the class $/ensuremath{/mathcal{SB}}$ of all short exact sequences $/ensuremath{/xymatrix{0 & A /ar[r]^{/alpha} & B /ar[r] & C /ar[r] & 0}}$ where $/Im /alpha$ has a supplement $V$ in $B$ and $V/cap /Im /alpha$ is bounded. The corresponding elements of $/Ext_{R}(C,A)$ are called $/beta$-elements. Over a noetherian integral domain of Krull dimension 1, $/beta$ elements form a proper class. In the category $/mathcal{T}_{R}$ over a Dedekind domain $R$, $/ensuremath{/mathcal{SB}}$ is a proper class; there are no nonzero $/ensuremath{/mathcal{SB}}$-projective $R$-modules and $/ensuremath{/mathcal{SB}}$-injective $R$-modules are only the injective $R$-modules. In the category $/mathcal{T}_{R}$, reduced $/ensuremath{/mathcal{SB}}$-coinjective $R$-modules are bounded $R$-modules.