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modules_over_a_ring [2010/07/29 15:46] (current)
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 +=====Modules over a ring=====
 +
 +Abbreviation: **RMod**
 +====Definition====
 +A \emph{module over a [[rings with identity]]} $\mathbf{R}$ is a structure $\mathbf{A}=\langle A,+,-,0,f_r\ (r\in R)\rangle$ such that
 +
 +
 +$\langle A,+,-,0\rangle $ is an [[abelian groups]]
 +
 +
 +$f_r$ preserves addition:  
 +$f_r(x+y)=f_r(x)+f_r(y)$
 +
 +
 +$f_{1}$ is the identity map:  $f_{1}(x)=x$
 +
 +
 +$f_{r+s}(x))=f_r(x)+f_s(x)$
 +
 +
 +$f_{r\circ s}(x)=f_r(f_s(x))$
 +
 +Remark:
 +$f_r$ is called \emph{scalar multiplication by $r$}, and $f_r(x)$ is usually written simply as $rx$.
 +
 +==Morphisms==
 +Let $\mathbf{A}$ and $\mathbf{B}$ be modules over a ring $\mathbf{R}$.
 +A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a group homomorphism and preserves all $f_r$:
 +
 +$h(f_r(x))=f_r(h(x))$
 +====Examples====
 +Example 1:
 +
 +====Basic results====
 +
 +
 +====Properties====
 +^[[Classtype]]  |variety |
 +^[[Equational theory]]  | |
 +^[[Quasiequational theory]]  | |
 +^[[First-order theory]]  | |
 +^[[Locally finite]]  |no |
 +^[[Residual size]]  |unbounded |
 +^[[Congruence distributive]]  |no |
 +^[[Congruence modular]]  |yes |
 +^[[Congruence n-permutable]]  |yes, $n=2$ |
 +^[[Congruence regular]]  |yes |
 +^[[Congruence uniform]]  |yes |
 +^[[Congruence extension property]]  |yes |
 +^[[Definable principal congruences]]  |no |
 +^[[Equationally def. pr. cong.]]  |no |
 +^[[Amalgamation property]]  | |
 +^[[Strong amalgamation property]]  | |
 +^[[Epimorphisms are surjective]]  | |
 +====Finite members====
 +
 +$\begin{array}{lr}
 +f(1)= &1\\
 +f(2)= &\\
 +f(3)= &\\
 +f(4)= &\\
 +f(5)= &\\
 +f(6)= &\\
 +\end{array}$
 +
 +====Subclasses====
 +====Superclasses====
 +[[Abelian groups]]
 +
 +
 +====References====
 +
 +[(Ln19xx>
 +)]
 +
 +
 +
 +
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 +