S-Semiprime Submodules and <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"> <a:mi>S</a:mi> </a:math>-Reduced Modules

This article introduces the concept of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M2"> <a:mi>S</a:mi> </a:math> -semiprime submodules which are a generalization of semiprime submodules and <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" id="M3"> <c:mi>S</c:mi> </c:math> -prime submodules. Let <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" id="M4"> <e:mi>M</e:mi> </e:math> be a nonzero unital R-module, where <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" id="M5"> <g:mi>R</g:mi> </g:math> is a commutative ring with a nonzero identity. Suppose that <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" id="M6"> <i:mi>S</i:mi> </i:math> is a multiplicatively closed subset of <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" id="M7"> <k:mi>R</k:mi> </k:math> . A submodule <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" id="M8"> <m:mi>P</m:mi> </m:math> of <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" id="M9"> <o:mi>M</o:mi> </o:math> is said to be an <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" id="M10"> <q:mi>S</q:mi> </q:math> -semiprime submodule if there exists a fixed <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" id="M11"> <s:mi>s</s:mi> <s:mo>∈</s:mo> <s:mi>S</s:mi> </s:math> , and whenever <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" id="M12"> <u:msup> <u:mrow> <u:mi>r</u:mi> </u:mrow> <u:mrow> <u:mi>n</u:mi> </u:mrow> </u:msup> <u:mi>m</u:mi> <u:mo>∈</u:mo> <u:mi>P</u:mi> </u:math> for some <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" id="M13"> <w:mi>r</w:mi> <w:mo>∈</w:mo> <w:mi>R</w:mi> <w:mo>,</w:mo> <w:mi>m</w:mi> <w:mo>∈</w:mo> <w:mi>M</w:mi> </w:math> , and <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" id="M14"> <y:mi>n</y:mi> <y:mo>∈</y:mo> <y:mi>ℕ</y:mi> </y:math> , then <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" id="M15"> <ab:mtext>srm</ab:mtext> <ab:mo>∈</ab:mo> <ab:mi>P</ab:mi> </ab:math> . Also, <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" id="M16"> <cb:mi>M</cb:mi> </cb:math> is said to be an <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" id="M17"> <eb:mi>S</eb:mi> </eb:math> -reduced module if there exists (fixed) <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" id="M18"> <gb:mi>s</gb:mi> <gb:mo>∈</gb:mo> <gb:mi>S</gb:mi> </gb:math> , and whenever <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" id="M19"> <ib:msup> <ib:mrow> <ib:mi>r</ib:mi> </ib:mrow> <ib:mrow> <ib:mi>n</ib:mi> </ib:mrow> </ib:msup> <ib:mi>m</ib:mi> <ib:mo>=</ib:mo> <ib:mn>0</ib:mn> </ib:math> for some <kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML" id="M20"> <kb:mi>r</kb:mi> <kb:mo>∈</kb:mo> <kb:mi>R</kb:mi> <kb:mo>,</kb:mo> <kb:mi>m</kb:mi> <kb:mo>∈</kb:mo> <kb:mi>M</kb:mi> </kb:math> , and <mb:math xmlns:mb="http://www.w3.org/1998/Math/MathML" id="M21"> <mb:mi>n</mb:mi> <mb:mo>∈</mb:mo> <mb:mi>ℕ</mb:mi> <mb:mo>,</mb:mo> </mb:math> then <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML" id="M22"> <ob:mtext>srm</ob:mtext> <ob:mo>=</ob:mo> <ob:mn>0</ob:mn> </ob:math> . In addition, to give many examples and characterizations of <qb:math xmlns:qb="http://www.w3.org/1998/Math/MathML" id="M23"> <qb:mi>S</qb:mi> </qb:math> -semiprime submodules and <sb:math xmlns:sb="http://www.w3.org/1998/Math/MathML" id="M24"> <sb:mi>S</sb:mi> </sb:math> -reduced modules, we characterize a certain class of semiprime submodules and reduced modules in terms of these concepts.