Almost Jordan rings

It is well known that any Jordan ring satisfies the identity: <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 left-parenthesis left-parenthesis a x right-parenthesis x right-parenthesis x plus a left-parenthesis left-parenthesis x x right-parenthesis x right-parenthesis equals 3 left-parenthesis a left-parenthesis x x right-parenthesis right-parenthesis x"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo stretchy="false">(</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>a</mml:mi> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi>x</mml:mi> <mml:mo>+</mml:mo> <mml:mi>a</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy="false">(</mml:mo> <mml:mi>a</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> <mml:mi>x</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">2((ax)x)x + a((xx)x) = 3(a(xx))x</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We show that this identity along with commutativity implies the Jordan identity in any semiprime ring. The proof requires characteistic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="not-equals 2 comma 3"> <mml:semantics> <mml:mrow> <mml:mo>≠</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\ne 2,3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.

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