Motivation: Let two algebra $A,C$ satisfy the property P and we have a short exact sequence $0\to A\to B\to C\to 0$ does it implies that B satisfies P too?

@Gro-Tsen let X be a topological space $X=G \sqcup F$ $G$ open $F$ closed such that both $G, F$ satisfy super Baire. Does this imply that $X$ is a super Baire

@DavidGao One another points with classical formulation but motivation from extension theory in algebra:

@DavidGao So contrary to your opinion I think the commutative Banach algebraic setting is interesting

@DavidGao from the view of dimension theory one may think of possible low dimensional examples of Banach algebra with desired property. I mean dimension in terms of tsr and real rank(M. Riefell's work)

@Gro-Tsen What can be said about the topological dimension of spaces you examined(The space with super Baire property)?Are there plenty of low dimensional cases?

. So a possible remedy is consideration of commutative Banach algebras

@DavidGao Regarding commutative Banach algebra I would suggest the following point: let A and B be two Banach algebras with super Baire property(In terms of essential ideal intersection). Is it true to say that every kind of tensor product $A\otimes B$ is a super baire algebra? This is inspired by that fact that the product of two compact baire space is again a baire space. On the other hand we pose this question in commutative C^* algebras we have an obvious "Yes" but if we drope the commutativity we have diffoculty with two sided ideals...

@DavidGao One may keep commutativity but drop the involution $*$ . That is "Banach algebras"

@DavidGao Since a manifold do not satisfy the super Baire property so it would be interesting to examin an appropriate NC analogy of the super Baire in an spectral triple

@Gro-Tsen May I ask you to add some tags as NCG or operator algebras or functional analysis. This is apparently unrealated to your question but in reality it is related to the subject

@YemonChoi "All you have done is a reformulated..." Ok his question is :"Is there a name for this property? So Yemon what is the job of a dictionary translating a name or a word into another language. So I do not see why is not a relevant answer to (the first edition of ) the OP question. Moreover after that I post the answer I found some papers who concern if an intersection of essential ideals is an essential ideal or not. (In case of rings)

Me too I do not know how or why it disappeared