Example of a ring of finite global dimension with flat qu0tient

I've been thinking about this for quite a while but I cannot seem to find an example of

If $k$ is a commutative ring of finite global dimension and I'm looking for a strictly not-commutative $k$-algebra $A$, which is $k$-flat and there is an ideal $I$ in $A$ such that $A/I$ is a regular commutative ring which is $k$-flat also....

Edit: and I is generated by an element which is not a unit not a zero divisor and it commutes with all the elements of A...

(the flatness condition seems to really restrict my possibilities....)

• Can't you just take $A$ to be the algebra of upper triangular $2\times 2$ matrices over $k$, and $I$ the ideal consisting of matrices whose first column is zero? Then $A$ is not only $k$-flat, but free, and $A/I\cong k$. – Jeremy Rickard Aug 7 '14 at 12:38
• I doubt this is possible with the flatness condition... I'll have to check the details later... – AIM_BLB Aug 9 '14 at 1:31
• "regular commutative" means "commutative"? Or you are asking for a particular version of "regular"? And your edit precludes Artinian rings from consideration... maybe that's what was interfering? – rschwieb Aug 11 '14 at 12:52
• it means commutative and regular – AIM_BLB Aug 11 '14 at 16:53