document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=867410>Question 867410</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1)Prove that if a+a=a a=0 a is an element of ring R\r<BR>\n" );
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document.write( "2) if every element of a ring R is an idempotent , prove that a+a=0 3) what is addition and multiplication table for {0,1,x,1+x} note that each element is an idempotent . <BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #523037 by </B><A HREF=/tutors/aboutme.mpl?userid=rothauserc><B>rothauserc(4718)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=rothauserc><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=rothauserc><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=523037\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 1)The properties of a Ring include that it is a Group under addition whose identity is 0 and inverse notation is -r where r is member of the Group.<BR>\n" );
document.write( "In a Group under addition every element (a) has an inverse such that<BR>\n" );
document.write( "a + b = b + a = 0 and b is a's inverse denoted by -a<BR>\n" );
document.write( "we are given a + a = 0, a = 0 which implies that a is its own inverse<BR>\n" );
document.write( "note that 0, the additive identity element is unique and a required member of the additive Group, under closure property a + a is a member of the additive Group<BR>\n" );
document.write( "2)An idempotent element a of the Ring is defined as<BR>\n" );
document.write( "a^2 = a^1, thus the element is idempotent under the Ring's multiplication and moreover we can say a^1 = a^2 = a^3 = a^4 = .....a^n or any positive integer n<BR>\n" );
document.write( "if every element of a ring R is an idempotent, we are dealing with a Boolean Ring<BR>\n" );
document.write( "a+a = (a+a)^2 = a^2 + a^2 +a^2 +a^2 = a +a +a +a<BR>\n" );
document.write( "now subtract a+a from both sides of the =<BR>\n" );
document.write( "0 = a +a<BR>\n" );
document.write( "addition table<BR>\n" );
document.write( "row/ 0   1   x   1+x<BR>\n" );
document.write( "column      <BR>\n" );
document.write( "0    0   1   x   1\r<BR>\n" );
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document.write( "1    1   0  1+x  x\r<BR>\n" );
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document.write( "x    x  1+x  0   1\r<BR>\n" );
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document.write( "1+x  1+x  x   1   0<BR>\n" );
document.write( "multiplication table<BR>\n" );
document.write( "row/ 0   1   x   1+x<BR>\n" );
document.write( "column     <BR>\n" );
document.write( "0    0   0   0   0  \r<BR>\n" );
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document.write( "1    0   1   x   1+x\r<BR>\n" );
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document.write( "x    0   x   x    0\r<BR>\n" );
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document.write( "1+x  0   x   0   1+x\r<BR>\n" );
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