@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .

psr: a skos:ConceptScheme .
psr:-D681HJ5Q-G
  skos:prefLabel "anneau commutatif"@fr, "commutative ring"@en ;
  a skos:Concept ;
  skos:narrower psr:-GTBC87Q1-Q .

psr:-GTBC87Q1-Q
  skos:definition """<p>In algebra, a commutative ring <i>R</i> is said to be <b>arithmetical</b> (or <b>arithmetic</b>) if any of the following equivalent conditions hold:
</p>
<ol><li>The localization <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\\\\displaystyle R_{\\\\mathfrak {m}}}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <msub>
          <mi>R</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mrow class="MJX-TeXAtom-ORD">
              <mi mathvariant="fraktur">m</mi>
            </mrow>
          </mrow>
        </msub>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\\\\displaystyle R_{\\\\mathfrak {m}}}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e92c94b739d631e31c2683d48a53204bab087646" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.257ex; height:2.509ex;" alt="R_{\\\\mathfrak {m}}"></span> of <i>R</i> at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\\\\displaystyle {\\\\mathfrak {m}}}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <mrow class="MJX-TeXAtom-ORD">
            <mi mathvariant="fraktur">m</mi>
          </mrow>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\\\\displaystyle {\\\\mathfrak {m}}}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adc0e9162e96758157a34a6e44967288b481a7cd" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:1.676ex;" alt="{\\\\mathfrak {m}}"></span> is a uniserial ring for every maximal ideal <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\\\\displaystyle {\\\\mathfrak {m}}}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <mrow class="MJX-TeXAtom-ORD">
            <mi mathvariant="fraktur">m</mi>
          </mrow>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\\\\displaystyle {\\\\mathfrak {m}}}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adc0e9162e96758157a34a6e44967288b481a7cd" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:1.676ex;" alt="{\\\\mathfrak {m}}"></span> of <i>R</i>.</li>
<li>For all ideals <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\\\\displaystyle {\\\\mathfrak {a}},{\\\\mathfrak {b}}}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <mrow class="MJX-TeXAtom-ORD">
            <mi mathvariant="fraktur">a</mi>
          </mrow>
        </mrow>
        <mo>,</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mrow class="MJX-TeXAtom-ORD">
            <mi mathvariant="fraktur">b</mi>
          </mrow>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\\\\displaystyle {\\\\mathfrak {a}},{\\\\mathfrak {b}}}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84b7742bed6d8d23c026def725678bf71ab31f99" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.389ex; height:2.509ex;" alt="{\\\\mathfrak  {a}},{\\\\mathfrak  {b}}"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\\\\displaystyle {\\\\mathfrak {c}}}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <mrow class="MJX-TeXAtom-ORD">
            <mi mathvariant="fraktur">c</mi>
          </mrow>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\\\\displaystyle {\\\\mathfrak {c}}}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b21924b960341255be18e538e51404718f29cbc0" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.905ex; height:1.676ex;" alt="{\\\\mathfrak {c}}"></span>,
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\\\\displaystyle {\\\\mathfrak {a}}\\\\cap ({\\\\mathfrak {b}}+{\\\\mathfrak {c}})=({\\\\mathfrak {a}}\\\\cap {\\\\mathfrak {b}})+({\\\\mathfrak {a}}\\\\cap {\\\\mathfrak {c}})}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <mrow class="MJX-TeXAtom-ORD">
            <mi mathvariant="fraktur">a</mi>
          </mrow>
        </mrow>
        <mo>∩<!-- ∩ --></mo>
        <mo stretchy="false">(</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mrow class="MJX-TeXAtom-ORD">
            <mi mathvariant="fraktur">b</mi>
          </mrow>
        </mrow>
        <mo>+</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mrow class="MJX-TeXAtom-ORD">
            <mi mathvariant="fraktur">c</mi>
          </mrow>
        </mrow>
        <mo stretchy="false">)</mo>
        <mo>=</mo>
        <mo stretchy="false">(</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mrow class="MJX-TeXAtom-ORD">
            <mi mathvariant="fraktur">a</mi>
          </mrow>
        </mrow>
        <mo>∩<!-- ∩ --></mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mrow class="MJX-TeXAtom-ORD">
            <mi mathvariant="fraktur">b</mi>
          </mrow>
        </mrow>
        <mo stretchy="false">)</mo>
        <mo>+</mo>
        <mo stretchy="false">(</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mrow class="MJX-TeXAtom-ORD">
            <mi mathvariant="fraktur">a</mi>
          </mrow>
        </mrow>
        <mo>∩<!-- ∩ --></mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mrow class="MJX-TeXAtom-ORD">
            <mi mathvariant="fraktur">c</mi>
          </mrow>
        </mrow>
        <mo stretchy="false">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\\\\displaystyle {\\\\mathfrak {a}}\\\\cap ({\\\\mathfrak {b}}+{\\\\mathfrak {c}})=({\\\\mathfrak {a}}\\\\cap {\\\\mathfrak {b}})+({\\\\mathfrak {a}}\\\\cap {\\\\mathfrak {c}})}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79be33a12b449a3467ca28cbccb812213703d90b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.637ex; height:2.843ex;" alt="{\\\\mathfrak  {a}}\\\\cap ({\\\\mathfrak  {b}}+{\\\\mathfrak  {c}})=({\\\\mathfrak  {a}}\\\\cap {\\\\mathfrak  {b}})+({\\\\mathfrak  {a}}\\\\cap {\\\\mathfrak  {c}})"></span></dd></dl></li>
<li>For all ideals <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\\\\displaystyle {\\\\mathfrak {a}},{\\\\mathfrak {b}}}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <mrow class="MJX-TeXAtom-ORD">
            <mi mathvariant="fraktur">a</mi>
          </mrow>
        </mrow>
        <mo>,</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mrow class="MJX-TeXAtom-ORD">
            <mi mathvariant="fraktur">b</mi>
          </mrow>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\\\\displaystyle {\\\\mathfrak {a}},{\\\\mathfrak {b}}}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84b7742bed6d8d23c026def725678bf71ab31f99" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.389ex; height:2.509ex;" alt="{\\\\mathfrak  {a}},{\\\\mathfrak  {b}}"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\\\\displaystyle {\\\\mathfrak {c}}}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <mrow class="MJX-TeXAtom-ORD">
            <mi mathvariant="fraktur">c</mi>
          </mrow>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\\\\displaystyle {\\\\mathfrak {c}}}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b21924b960341255be18e538e51404718f29cbc0" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.905ex; height:1.676ex;" alt="{\\\\mathfrak {c}}"></span>,
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\\\\displaystyle {\\\\mathfrak {a}}+({\\\\mathfrak {b}}\\\\cap {\\\\mathfrak {c}})=({\\\\mathfrak {a}}+{\\\\mathfrak {b}})\\\\cap ({\\\\mathfrak {a}}+{\\\\mathfrak {c}})}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <mrow class="MJX-TeXAtom-ORD">
            <mi mathvariant="fraktur">a</mi>
          </mrow>
        </mrow>
        <mo>+</mo>
        <mo stretchy="false">(</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mrow class="MJX-TeXAtom-ORD">
            <mi mathvariant="fraktur">b</mi>
          </mrow>
        </mrow>
        <mo>∩<!-- ∩ --></mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mrow class="MJX-TeXAtom-ORD">
            <mi mathvariant="fraktur">c</mi>
          </mrow>
        </mrow>
        <mo stretchy="false">)</mo>
        <mo>=</mo>
        <mo stretchy="false">(</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mrow class="MJX-TeXAtom-ORD">
            <mi mathvariant="fraktur">a</mi>
          </mrow>
        </mrow>
        <mo>+</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mrow class="MJX-TeXAtom-ORD">
            <mi mathvariant="fraktur">b</mi>
          </mrow>
        </mrow>
        <mo stretchy="false">)</mo>
        <mo>∩<!-- ∩ --></mo>
        <mo stretchy="false">(</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mrow class="MJX-TeXAtom-ORD">
            <mi mathvariant="fraktur">a</mi>
          </mrow>
        </mrow>
        <mo>+</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mrow class="MJX-TeXAtom-ORD">
            <mi mathvariant="fraktur">c</mi>
          </mrow>
        </mrow>
        <mo stretchy="false">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\\\\displaystyle {\\\\mathfrak {a}}+({\\\\mathfrak {b}}\\\\cap {\\\\mathfrak {c}})=({\\\\mathfrak {a}}+{\\\\mathfrak {b}})\\\\cap ({\\\\mathfrak {a}}+{\\\\mathfrak {c}})}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef983793edbe0afc72260628a49f6611544a538a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.895ex; height:2.843ex;" alt="{\\\\mathfrak  {a}}+({\\\\mathfrak  {b}}\\\\cap {\\\\mathfrak  {c}})=({\\\\mathfrak  {a}}+{\\\\mathfrak  {b}})\\\\cap ({\\\\mathfrak  {a}}+{\\\\mathfrak  {c}})"></span></dd></dl></li></ol>
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Arithmetical_ring">https://en.wikipedia.org/wiki/Arithmetical_ring</a>)"""@en ;
  a skos:Concept ;
  skos:inScheme psr: ;
  skos:broader psr:-D681HJ5Q-G ;
  skos:prefLabel "anneau arithmétique"@fr, "arithmetic ring"@en ;
  skos:altLabel "arithmetical ring"@en ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Arithmetical_ring> .