# Differences

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+ | =====Integral Domain===== | ||

+ | Abbreviation: **IntDom** | ||

+ | ====Definition==== | ||

+ | An \emph{integral domain} is a [[commutative rings with identity]] $\mathbf{R}=\langle R,+,-,0,\cdot,1\rangle$ that | ||

+ | |||

+ | |||

+ | has no zero divisors: | ||

+ | $\forall x,y\ (x\cdot y=0\Longrightarrow x=0\ \mbox{or}\ y=0)$ | ||

+ | |||

+ | ==Morphisms== | ||

+ | Let $\mathbf{R}$ and $\mathbf{S}$ be integral domains. A morphism from $\mathbf{R}$ | ||

+ | to $\mathbf{S}$ is a function $h:R\rightarrow S$ that is a homomorphism: | ||

+ | |||

+ | $h(x+y)=h(x)+h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(1)=1$ | ||

+ | |||

+ | Remark: | ||

+ | It follows that $h(0)=0$ and $h(-x)=-h(x)$. | ||

+ | |||

+ | ====Examples==== | ||

+ | Example 1: $\langle\mathbb{Z},+,-,0,\cdot,1\rangle$, the ring of integers with addition, subtraction, zero, and multiplication is an integral domain. | ||

+ | |||

+ | |||

+ | ====Basic results==== | ||

+ | Every finite integral domain is a [[fields]]. | ||

+ | |||

+ | ====Properties==== | ||

+ | ^[[Classtype]] |Universal class | | ||

+ | ^[[Equational theory]] | | | ||

+ | ^[[Quasiequational theory]] | | | ||

+ | ^[[First-order theory]] | | | ||

+ | ^[[Locally finite]] | | | ||

+ | ^[[Residual size]] | | | ||

+ | ^[[Congruence distributive]] | | | ||

+ | ^[[Congruence modular]] |yes | | ||

+ | ^[[Congruence n-permutable]] |yes, $n=2$ | | ||

+ | ^[[Congruence regular]] |yes | | ||

+ | ^[[Congruence uniform]] |yes | | ||

+ | ^[[Congruence extension property]] | | | ||

+ | ^[[Definable principal congruences]] | | | ||

+ | ^[[Equationally def. pr. cong.]] | | | ||

+ | ^[[Amalgamation property]] | | | ||

+ | ^[[Strong amalgamation property]] | | | ||

+ | ^[[Epimorphisms are surjective]] | | | ||

+ | ====Finite members==== | ||

+ | |||

+ | $\begin{array}{lr} | ||

+ | f(1)= &1\\ | ||

+ | f(2)= &1\\ | ||

+ | f(3)= &1\\ | ||

+ | f(4)= &1\\ | ||

+ | f(5)= &1\\ | ||

+ | f(6)= &0\\ | ||

+ | \end{array}$ | ||

+ | |||

+ | ====Subclasses==== | ||

+ | [[Unique factorization domains]] | ||

+ | |||

+ | ====Superclasses==== | ||

+ | [[Commutative rings with identity]] | ||

+ | |||

+ | |||

+ | ====References==== | ||

+ | |||

+ | [(Ln19xx> | ||

+ | )] |

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