Strong commutativity preserving maps on Lie ideals

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• 作者：Jer-Shyong Lin ; Cheng-Kai Liu
• 关键词：Prime ring ; Lie ideal ; Strong commutativity preserving
• 刊名：Linear Algebra and its Applications
• 出版年：2008
• 出版时间：1 April 2008
• 年：2008
• 卷：428
• 期：7
• 页码：1601-1609
• 全文大小：137 K

文摘

Let che/MiamiImageURL/B6V0R-4R7F4GT-4-2/0?wchp=dGLbVzb-zSkWb"" alt=""Click to view the MathML source"" align=""absbottom"" border=""0"" height=13 width=18> be a prime ring and let che/MiamiImageURL/B6V0R-4R7F4GT-4-F/0?wchp=dGLbVzb-zSkWb"" alt=""Click to view the MathML source"" align=""absbottom"" border=""0"" height=13 width=15> be a noncentral Lie ideal of che/MiamiImageURL/B6V0R-4R7F4GT-4-G/0?wchp=dGLbVzb-zSkWb"" alt=""Click to view the MathML source"" align=""absbottom"" border=""0"" height=13 width=18>. An additive map che/MiamiImageURL/B6V0R-4R7F4GT-4-H/0?wchp=dGLbVzb-zSkWb"" alt=""Click to view the MathML source"" align=""absbottom"" border=""0"" height=16 width=81> is called strong commutativity preserving (SCP) on che/MiamiImageURL/B6V0R-4R7F4GT-4-J/0?wchp=dGLbVzb-zSkWb"" alt=""Click to view the MathML source"" align=""absbottom"" border=""0"" height=13 width=15> if [f(x),f(y)]=[x,y] for all che/MiamiImageURL/B6V0R-4R7F4GT-4-M/0?wchp=dGLbVzb-zSkWb"" alt=""Click to view the MathML source"" align=""absbottom"" border=""0"" height=16 width=57>. In this paper we show that if f is SCP on che/MiamiImageURL/B6V0R-4R7F4GT-4-N/0?wchp=dGLbVzb-zSkWb"" alt=""Click to view the MathML source"" align=""absbottom"" border=""0"" height=13 width=15>, then there exist che/MiamiImageURL/B6V0R-4R7F4GT-4-P/0?wchp=dGLbVzb-zSkWb"" alt=""Click to view the MathML source"" align=""absbottom"" border=""0"" height=13 width=41>, λ²=1 and an additive map che/MiamiImageURL/B6V0R-4R7F4GT-4-4/0?wchp=dGLbVzb-zSkWb"" alt=""Click to view the MathML source"" align=""absbottom"" border=""0"" height=17 width=104> such that f(x)=λx+μ(x) for all che/MiamiImageURL/B6V0R-4R7F4GT-4-6/0?wchp=dGLbVzb-zSkWb"" alt=""Click to view the MathML source"" align=""absbottom"" border=""0"" height=13 width=43> where che/MiamiImageURL/B6V0R-4R7F4GT-4-7/0?wchp=dGLbVzb-zSkWb"" alt=""Click to view the MathML source"" align=""absbottom"" border=""0"" height=13 width=14> is the extended centroid of che/MiamiImageURL/B6V0R-4R7F4GT-4-8/0?wchp=dGLbVzb-zSkWb"" alt=""Click to view the MathML source"" align=""absbottom"" border=""0"" height=13 width=18>, unless charche/MiamiImageURL/B6V0R-4R7F4GT-4-9/0?wchp=dGLbVzb-zSkWb"" alt=""Click to view the MathML source"" align=""absbottom"" border=""0"" height=13 width=47> and 8057229e1deca"">che/MiamiImageURL/B6V0R-4R7F4GT-4-B/0?wchp=dGLbVzb-zSkWb"" alt=""Click to view the MathML source"" align=""absbottom"" border=""0"" height=13 width=18> satisfies the standard identity of degree 4.