# Differences

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+ | =====Commutative rings with identity===== | ||

+ | Abbreviation: **CRng$_1$** | ||

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

+ | A \emph{commutative ring with identity} is a [[rings with identity]] $\mathbf{R}=\langle R,+,-,0,\cdot,1 | ||

+ | \rangle$ such that | ||

+ | $\cdot$ is commutative: $x\cdot y=y\cdot x$ | ||

+ | |||

+ | ==Morphisms== | ||

+ | Let $\mathbf{R}$ and $\mathbf{S}$ be commutative rings with identity. 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, multiplication, and one. | ||

+ | |||

+ | |||

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

+ | $0$ is a zero for $\cdot$: $0\cdot x=x$ and $x\cdot 0=0$. | ||

+ | |||

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

+ | ^[[Classtype]] |variety | | ||

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

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

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

+ | ^[[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]] | | | ||

+ | ^[[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)= &4\\ | ||

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

+ | f(6)= &1\\ | ||

+ | \end{array}$ | ||

+ | |||

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

+ | [[Boolean algebras]] | ||

+ | |||

+ | [[Integral domains]] | ||

+ | |||

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

+ | [[Commutative rings]] | ||

+ | |||

+ | [[Rings with identity]] | ||

+ | |||

+ | |||

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

+ | |||

+ | [(Ln19xx> | ||

+ | )] |

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