Let $A$ be a commutative ring and endow the closed subsets of $\mathrm{Spec}(A)$ with the Grothendieck topology of finite covers. One may ask if the presheaf $V \mapsto A/I(V)$ is a sheaf. This is not true in general and is related (but not equivalent) to the following pure algebraic question:

In which commutative rings $A$ are the radical ideals closed under sum?

The property can be checked locally. It holds in dimension $0$, and also for integral domains of dimension 1. It doesn't hold for the $2$-dimensional ring $k[x,y]$ (consider $(x^2 + y)+(y) = (x^2,y)$), nor for the 1-dimensional ring $\bigl(k[x,y]/(x^2 y + y^2)\bigr)_{(x,y)}$.

Are there other interesting examples/counterexamples or approaches for a general classification? I think the property has some algebro-geometric interpretation: All intersections of closed subschemes are transversal. See also SE/322872.

Let $A$ be a commutative ring and endow the closed subsets of $\operatorname{Spec}(A)$ with the Grothendieck topology of finite covers. One may ask if the presheaf $V \mapsto A/I(V)$ is a sheaf. This is not true in general and is related (but not equivalent) to the following pure algebraic question:

In which commutative rings $A$ are the radical ideals closed under sum?

The property can be checked locally. It holds in dimension $0$, and also for integral domains of dimension 1. It doesn't hold for the $2$-dimensional ring $k[x,y]$ (consider $(x^2 + y)+(y) = (x^2,y)$), nor for the $1$-dimensional ring $\bigl(k[x,y]/(x^2 y + y^2)\bigr)_{(x,y)}$.

Are there other interesting examples/counterexamples or approaches for a general classification? I think the property has some algebro-geometric interpretation: All intersections of closed subschemes are transversal. See also Conditions for $\sqrt{\mathfrak{a + b}} = \sqrt{\mathfrak{a}} + \sqrt{\mathfrak{b}}$.

Let $A$ be a commutative ring and endow the closed subsets of $\mathrm{Spec}(A)$ with the Grothendieck topology of finite covers. One may ask if the presheaf $V \mapsto A/I(V)$ is a sheaf. This is not true in general and is related (but not equivalent) to the following pure algebraic question:

In which commutative rings $A$ are the radical ideals closed under sum?

The property can be checked locally. It holds in dimension $0$, and also for integral domains of dimension 1. It doesn't hold for the $2$-dimensional ring $k[x,y]$ (consider $(x^2 + y)+(y) = (x^2,y)$), nor for the 1-dimensional ring $\bigl(k[x,y]/(x^2 y + y^2)\bigr)_{(x,y)}$.

Are there other interesting examples/counterexamples or approaches for a general classification? I think the property has some algebro-geometric interpretation: All intersections of closed subschemes are transversal. See also SE/322872.

Let $A$ be a commutative ring and endow the closed subsets of $\mathrm{Spec}(A)$ with the Grothendieck topology of finite covers. One may ask if the presheaf $V \mapsto A/I(V)$ is a sheaf. This is not true in general and is related (but not equivalent) to the following pure algebraic question:

In which commutative rings $A$ are the radical ideals closed under sum?

The property can be checked locally. It holds in dimension $0$, and also for integral domains of dimension 1. It doesn't hold for the $2$-dimensional ring $k[x,y]$ (consider $(x^2 + y)+(y) = (x^2,y)$), nor for the 1-dimensional ring $\bigl(k[x,y]/(x^2 y + y^2)\bigr)_{(x,y)}$.

Are there other interesting examples/counterexamples or approaches for a general classification? I think the property has some algebro-geometric interpretation: All intersections of closed subschemes are transversal. See also SE/322872.

let $A$ be a ring and endow the closed subsets of $Spec A$ with the grothendieck topology of finite covers. it would be nice if $V \mapsto A/I(V)$ defines a sheaf. this is not true in general and is related (equivalent?) to the following pure algebraic question:

in which rings $A$ are the radical ideals closed under sum?

if $A$ is a noetherian integral domain of dimension $0$ or $1$, it's true. you cannot omit integral here (consider $k[x,y]/(x^2 y + y^2)$ localized at $(x,y)$), nor the dimension (in $k[x,y]$, consider $(x^2 + y)+(y) = (x^2,y)$). are there other interesting examples/counterexamples or approaches for general classification?

Let $A$ be a commutative ring and endow the closed subsets of $\mathrm{Spec}(A)$ with the Grothendieck topology of finite covers. One may ask if the presheaf $V \mapsto A/I(V)$ is a sheaf. This is not true in general and is related (but not equivalent) to the following pure algebraic question:

In which commutative rings $A$ are the radical ideals closed under sum?

The property can be checked locally. It holds in dimension $0$, and also for integral domains of dimension 1. It doesn't hold for the $2$-dimensional ring $k[x,y]$ (consider $(x^2 + y)+(y) = (x^2,y)$), nor for the 1-dimensional ring $\bigl(k[x,y]/(x^2 y + y^2)\bigr)_{(x,y)}$.

Are there other interesting examples/counterexamples or approaches for a general classification? I think the property has some algebro-geometric interpretation: All intersections of closed subschemes are transversal. See also SE/322872.