Mathematical Structures: Rings with identity

# Rings with identity

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http://mathcs.chapman.edu/structuresold/files/Rings_with_identity.pdf
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\newtheorem*{morphisms}{Morphisms}
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\begin{document}
\textbf{\Large Rings with identity}

\abbreviation{Rng$_1$}
\begin{definition}
A \emph{ring with identity} is a structure $\mathbf{R}=\langle R,+,-,0,\cdot,1 \rangle$ of type $\langle 2,1,0,2,0\rangle$ such that

$\langle R,+,-,0,\cdot\rangle$ is a \href{Rings.pdf}{ring}

$1$ is an identity for $\cdot$:  $x\cdot 1=x$, $1\cdot x=x$
\end{definition}

\begin{morphisms}
Let $\mathbf{R}$ and $\mathbf{S}$ be 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 $h(-x)=-h(x)$.

\end{morphisms}
\begin{basic_results}
$0$ is a zero for $\cdot$: $0\cdot x=0$ and $x\cdot 0=0$.
\end{basic_results}
\begin{examples}
\begin{example}
$\langle\mathbb{Z},+,-,0,\cdot,1\rangle$, the ring of integers with addition, subtraction, zero, multiplication, and one.
\end{example}
\end{examples}
\begin{table}[h]
\begin{properties} (\href{http://math.chapman.edu/cgi-bin/structures?Properties}{description})

\begin{tabular}{|ll|}\hline
Classtype & variety\\\hline
Equational theory & decidable\\\hline
Quasiequational theory & \\\hline
First-order theory & undecidable\\\hline
Locally finite & no\\\hline
Residual size & unbounded\\\hline
Congruence distributive & no\\\hline
Congruence modular & yes\\\hline
Congruence n-permutable & yes, $n=2$\\\hline
Congruence regular & yes\\\hline
Congruence uniform & yes\\\hline
Congruence extension property & \\\hline
Definable principal congruences & \\\hline
Equationally def. pr. cong. & \\\hline
Amalgamation property & \\\hline
Strong amalgamation property & \\\hline
Epimorphisms are surjective & \\\hline
\end{tabular}
\end{properties}
\end{table}
\begin{finite_members} $f(n)=$ number of members of size $n$.

$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &1\\ f(4)= &4\\ f(5)= &1\\ f(6)= &1\\ \end{array}$

\href{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A037291}{Finite rings with identity in the Encyclopedia of Integer Sequences}
\end{finite_members}

\hyperbaseurl{http://math.chapman.edu/structures/files/}
\begin{subclasses}\

\href{Commutative_rings_with_identity.pdf}{Commutative rings with identity}

\href{Division_rings.pdf}{Division rings}

\end{subclasses}
\begin{superclasses}\

\href{Rings.pdf}{Rings}

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

\end{thebibliography}

\end{document}
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