# Differences

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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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