In this paper, we define the free <mml:math alttext="$A$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>A</mml:mi></mml:math>-ring over <mml:math alttext="$K$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>K</mml:mi></mml:math> on a set <mml:math alttext="$X$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>X</mml:mi></mml:math>, categorically, and
parallel some results from the theory of free algebras. We show that the free <mml:math alttext="$A$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>A</mml:mi></mml:math>-ring over <mml:math alttext="$K$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>K</mml:mi></mml:math> on <mml:math alttext="$X$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>X</mml:mi></mml:math>,
denoted by <mml:math alttext="$A_K \left\{ X \right\}$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>A</mml:mi><mml:mi>K</mml:mi></mml:msub><mml:mrow><mml:mo>{</mml:mo><mml:mi>X</mml:mi><mml:mo>}</mml:mo></mml:mrow></mml:math>, is graded.

The free <mml:math alttext="$A$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>A</mml:mi></mml:math>-ring is a graded <mml:math alttext="$A$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>A</mml:mi></mml:math>-ring

The free <svg style="vertical-align:-0.0pt" height="11.4625" version="1.1" viewBox="0 0 12.5875 11.4625" width="12.5875" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,11.4)"><path id="x1D434" d="M716 28l-8 -28h-254l8 28q55 5 66 17.5t5 47.5l-30 145h-232l-74 -138q-21 -43 -11 -54.5t72 -17.5l-6 -28h-235l8 28q53 7 73.5 21t53.5 70l319 540l33 8q6 -43 33 -176l80 -381q10 -49 26.5 -63t72.5 -19zM495 281l-49 264h-2l-149 -264h200z" /></g></svg>-ring is a graded <svg style="vertical-align:-0.0pt" height="11.4625" version="1.1" viewBox="0 0 12.5875 11.4625" width="12.5875" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,11.4)"><use xlink:href="#x1D434"/></g></svg>-ring

International Journal of Mathematics and Mathematical Sciences

The free A-ring is a graded A-ring

Abstract

Copyright