document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=32844>Question 32844</A>: <I><SPAN STYLE=\"word-break: break-all;\"> (a) Prove that the set S of rational numbers (in lowest term) with odd denominators is<BR>\n" );
document.write( "a subring of Q.<BR>\n" );
document.write( "(b) Let I be the set of elements of S with even numerators. Prove that I is an ideal in<BR>\n" );
document.write( "S.<BR>\n" );
document.write( "(c) Show that S/I consists of exactly two elements.\r<BR>\n" );
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document.write( "Thank You </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #19368 by </B><A HREF=/tutors/aboutme.mpl?userid=venugopalramana><B>venugopalramana(3286)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=venugopalramana><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.geocities.com/venugopalramana/FREE_MATHS_ASSISTANCE.html valign=middle><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=19368\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  S IS THE SET OF RATIONAL NUMBERS WITH D.R. BEING ODD NUMBER  ..THAT<BR>\n" );
document.write( "IS ELEMENTS ARE OF THE TYPE P/(2Q-1)<BR>\n" );
document.write( "TST S IS A SUB RING OF Q.<BR>\n" );
document.write( "FOR THIS WE NEED TO SHOW<BR>\n" );
document.write( "1.IF A AND B ARE ELEMENTS OF S THEN A-B IS ALSO AN ELEMENT OF S.<BR>\n" );
document.write( "LET A=P1/(2Q1-1)...AND....B=P2/(2Q2-1)<BR>\n" );
document.write( "A-B={P1(2Q2-1)-P2(2Q1-1)}/(2Q1-1)(2Q2-1)...WE FIND THAT D.R IS PRODUCT<BR>\n" );
document.write( "OF 2 ODD NUMBERS AND HENCE IS ODD.SO THE RESULTANT VALUE OF A-B IS<BR>\n" );
document.write( "STILL IN THE FORM OF SOME P/(2Q-1)...SO THIS BELONGS TO S<BR>\n" );
document.write( "2.IF A AND B ARE ELEMENTS OF S THEN AB IS ALSO AN ELEMENT OF S<BR>\n" );
document.write( "WITH SAME A AND B AS ABOVE WE HAVE<BR>\n" );
document.write( "AB=(P1P2)/(2Q1-1)(2Q2-1)...WHICH IS AGAIN OF THE TYPE SOME P/(2Q-1)<BR>\n" );
document.write( "AND HENCE AN ELEMENT OF S<BR>\n" );
document.write( "THESE 2 CRITERIA BEING NECESSARY AND SUFFICIENT FOR S TO BE A SUB RING<BR>\n" );
document.write( "OF Q WE CONCLUDE THE RESULT.<BR>\n" );
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document.write( "I IS SET OF EVEN N.R....SO THE ELEMENTS ARE 2P/(2Q-1)..TYPE.<BR>\n" );
document.write( "TO SHOW THAT THIS IS AN IDEAL OF S.<BR>\n" );
document.write( "FOR THIS WE HAVE TO PROVE THAT<BR>\n" );
document.write( "1.IF A AND B ARE ELEMENTS OF I ,THEN A-B IS AN ELEMENT OF I.<BR>\n" );
document.write( "LET A =2P1/(2Q-1)....B=2P2/(2Q2-1)....<BR>\n" );
document.write( "A-B = 2[P1(2Q2-1)-P2(2Q1-1)]/(2Q1-1)(2Q2-1).....WE FIND THAT N.R IS A<BR>\n" );
document.write( "MULTIPLE OF 2 AND HENCE AN EVEN NUMBER.DENOMINATOR IS PRODUCT OF 2 ODD<BR>\n" );
document.write( "NUMBERS AND HENCE AN ODD NUMBER.<BR>\n" );
document.write( "HENCE A-B IS EQUAL TO SOME 2P/(2Q-1)...THAT IS IT BELONGS TO I<BR>\n" );
document.write( "2.IF A IS AN ELEMENT OF I AND R IS AN ELEMENT OF S THEN A*R AND R*A<BR>\n" );
document.write( "ARE ELEMENTS OF I.<BR>\n" );
document.write( "LET R=R1/(2Q3-1)....WE HAVE<BR>\n" );
document.write( "A*R=R*A=[(R1)(2P1)]/(2Q1-1)(2Q3-1)...WHICH IS AGAIN OF THE TYPE SOME<BR>\n" );
document.write( "2P/(2Q-1)..FOR THE SAME REASONS GIVEN ABOVE.<BR>\n" );
document.write( "THESE 2 CRITERIA BEING NECESSARY AND SUFFICIENT CRITERIA FOR I TO BE<BR>\n" );
document.write( "AN IDEAL OF S ,WE CONCLUDE THE RESULT.\r<BR>\n" );
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document.write( "BY DEFINITION S/I IS THE SET OF COSETS OF I IN S<BR>\n" );
document.write( "I/S={S+A:A IS AN ELEMENT OF S}<BR>\n" );
document.write( "NOW WE HAVE I AS THE SET OF ELEMENTS OF THE TYPE...EVEN /ODD NUMBER<BR>\n" );
document.write( "WHERE AS S IS THE SET OF ELEMENTS OF THE TYPE,....EVEN/ODD AND ODD/ODD<BR>\n" );
document.write( "ELEMENTS...SO WHEN YOU MAKE A COSET , YOU WILL HAVE ONLY 2 DISTINCT<BR>\n" );
document.write( "CLASSES ALWAYS NAMELY ODD AND EVEN IN NR.SINCE S/I IS THE SET OF<BR>\n" );
document.write( "DISTINCT COSETS ,WE HAVE ONLY 2 DISTINCT CLASSES OR ELEMENTS IN THIS<BR>\n" );
document.write( "QUOTIENT RING OR RESIDUAL CLASS AS IT IS CALLED.<BR>\n" );
document.write( "HENCE THERE ARE ONLY 2 DISTINCT ELEMENTS IN S/I,ONE CLASS OF ELEMENTS OS 2P/(2Q-1)AND ANOTHER CLASS OF ELEMENTS OF TYPE (2P-1)/(2Q-1)  </SPAN>\n" );
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