Serial module

In <a href="/facts/Abstract_algebra/YNNE1CNz">abstract algebra</a>, a uniserial module M is a <a href="/facts/Module_(mathematics)/jP0LIhFL">module</a> over a <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a> R, whose <a href="/facts/Submodule/jP0LIhFL">submodules</a> are <a href="/facts/Total_order/xnTtmuYJ">totally ordered</a> by <a href="/facts/Inclusion_(set_theory)/SJGERmbA">inclusion</a>. This means simply that for any two submodules N1 and N2 of M, either 
 
 
 
 
 N
 
 1
 
 
 ⊆
 
 N
 
 2
 
 
 
 
 {\displaystyle N_{1}\subseteq N_{2}}
 
 or 
 
 
 
 
 N
 
 2
 
 
 ⊆
 
 N
 
 1
 
 
 
 
 {\displaystyle N_{2}\subseteq N_{1}}
 
. A module is called a serial module if it is a <a href="/facts/Direct_sum_of_modules/Y6PBJnjg">direct sum</a> of uniserial modules. A ring R is called a right uniserial ring if it is uniserial as a right module over itself, and likewise called a right serial ring if it is a right serial module over itself. Left uniserial and left serial rings are defined in a similar way, and are in general distinct from their right-sided counterparts.
An easy motivating example is the <a href="/facts/Quotient_ring/X2WkXyEw">quotient ring</a> 
 
 
 
 
 Z
 
 
 /
 
 n
 
 Z
 
 
 
 {\displaystyle \mathbb {Z} /n\mathbb {Z} }
 
 for any <a href="/facts/Integer/PUwwrawx">integer</a> 
 
 
 
 n
 >
 1
 
 
 {\displaystyle n>1}
 
. This ring is always serial, and is uniserial when n is a <a href="/facts/Prime_power/R0vo3uYp">prime power</a>.
The term uniserial has been used differently from the above definition: for clarification see below.
A partial alphabetical list of important contributors to the theory of serial rings includes the mathematicians Keizo Asano, I. S. Cohen, <a href="/facts/Paul_Cohn/gnBlbFuK">P.M. Cohn</a>, Yu. Drozd, <a href="/facts/David_Eisenbud/hw2yEnnx">D. Eisenbud</a>, A. Facchini, <a href="/facts/Alfred_Goldie/WBpdWTRT">A.W. Goldie</a>, Phillip Griffith, <a href="/facts/Irving_Kaplansky/nRLyp7K9">I. Kaplansky</a>, V.V Kirichenko, <a href="/facts/Gottfried_K%C3%B6the/y7A72q5g">G. Köthe</a>, H. Kuppisch, I. Murase, <a href="/facts/Tadashi_Nakayama_(mathematician)/3yrSLWTm">T. Nakayama</a>, P. Příhoda, G. Puninski, and R. Warfield.
Following the common ring theoretic convention, if a left/right dependent condition is given without mention of a side (for example, uniserial, serial, <a href="/facts/Artinian_ring/BlzWdfTL">Artinian</a>, <a href="/facts/Noetherian/beQxPUnf">Noetherian</a>) then it is assumed the condition holds on both the left and right. Unless otherwise specified, each ring in this article is a <a href="/facts/Ring_with_unity/JnCUXz63">ring with unity</a>, and each module is <a href="/facts/Unital_module/jP0LIhFL">unital</a>.

Serial module open-in-new

Serial module