Still trying to get my head around certain foundational concepts in algebraic geometry here, so pray bear with me...
Is it true to say that if $R$ is a ring, then every (associative) algebra over $R$ does itself have the structure of a ring?
Furthermore, provided the above holds water, is it true to say that if $R$ is reduced, then any $R$-algebra, regarded as a ring, is also reduced?
I look forward to your responses.
Every major definition of an associative algebra $A$ over a (commutative) ring $R$ makes $A$ a ring.
Every ring is a $\mathbb Z$ algebra, and $\mathbb Z$ is reduced, so if algebras over reduced rings were reduced, all rings would be reduced.
As someone already gave in the comments, $R[x]/(x^2)$ is a simple example of an $R$ algebra that isn’t reduced, and $R$ can be any nonzero commutative ring, even a reduced one.