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<b>let </b><font color="Maroon" title="c1">R</font> be  <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a>;<br/><b>let </b><font color="Maroon" title="c2">I</font> be   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font>;<br/>



<b>thus </b><a NAME="E1:35"/>
( <font color="Maroon" title="c2">I</font> is <a href="ring_1.html#V2" title="RING_1:attr.2">prime</a> implies <font color="Maroon" title="c1">R</font> <a href="ring_1.html#NK3" title="RING_1:NK.3">/</a> <font color="Maroon" title="c2">I</font> is   <a href="vectsp_2.html#NM2" title="VECTSP_2:NM.2">domRing</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:35_1"/>
<font color="Maroon" title="c2">I</font> is <a href="ring_1.html#V2" title="RING_1:attr.2">prime</a>
 ;<br/>

<a NAME="E2:35_1"/><b>then </b>
( <font color="Maroon" title="c2">I</font> is <a href="ring_1.html#V1" title="RING_1:attr.1">quasi-prime</a> &amp; <font color="Maroon" title="c2">I</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a> )
 
<b>by </b><i><a class="ref" href="ring_1.html#D2" target="_self" title="RING_1:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E3:35_1"/>
<font color="Maroon" title="c1">R</font> <a href="ring_1.html#NK3" title="RING_1:NK.3">/</a> <font color="Maroon" title="c2">I</font> is   <a href="vectsp_2.html#NM2" title="VECTSP_2:NM.2">domRing</a>
 <b>by </b><i><a class="ref" href="ring_1.html#T16" target="_self" title="RING_1:th.16">Th16</a>, <a class="ref" href="ring_1.html#T17" target="_self" title="RING_1:th.17">Th17</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:35"/>
<font color="Maroon" title="c1">R</font> <a href="ring_1.html#NK3" title="RING_1:NK.3">/</a> <font color="Maroon" title="c2">I</font> is   <a href="vectsp_2.html#NM2" title="VECTSP_2:NM.2">domRing</a>
 ;<br/>

<b>hence </b><a NAME="E3:35"/>
( <font color="Maroon" title="c2">I</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a> &amp; <font color="Maroon" title="c2">I</font> is <a href="ring_1.html#V1" title="RING_1:attr.1">quasi-prime</a> )
 <b>by </b><i><a class="ref" href="ring_1.html#T16" target="_self" title="RING_1:th.16">Th16</a>, <a class="ref" href="ring_1.html#T17" target="_self" title="RING_1:th.17">Th17</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="ring_1.html#D2" target="_self" title="RING_1:def.2">RING_1:def 2</a><br/>


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