[A weird thought born when thinking about art x mathematics] In a rough sense, when an artist is making an artwork, the details of the artwork seemed to reflect the historical background, the artist's emotion and ideas onto it. Thus it seems it can be thought as the following:

Let the set S to contain all historical context, emotions and ideas. Then the making of an artwork seemed to act like a map that maps S to the artwork such that it has some form that is a subset of S. Therefore under this framework, it seems an artwork is some kind of injective map. It is then tempting to say this is…

Hi all! Take a look at this ring $A'=A[X_a]_{a\in A}/(a^2X_a-a,aX_a^2-X_a)_{a\in A}$, where $A$ is some ring (commutative with unity). I'm trying to prove that for a maximal ideal $\mathfrak{m}$ of $A'$ it holds that $A'_\mathfrak{m}$ (the localization at $\mathfrak{m}$) is a field.

It's enough to prove that the maximal ideal of $A'_\mathfrak{m}$ is $\{0\}$. So therefore I took an element $x\in\mathfrak{m}A'_\mathfrak{m}$ (the maximal ideal), and assumed that $x\neq 0$. If $x=\frac{a}{s}$ this means that there exists no z. div. of $a$ in $A'\setminus\mathfrak{m}$.