Which of the following structures controls the flow of materials in and out of the cell's nucleus?

The nuclear membrane, also known as the nuclear envelope, is a double-layered structure that surrounds the nucleus of a cell. Its primary function is to regulate the transportation of substances between the nucleus and the cytoplasm, thereby maintaining the distinct environment of the nucleus. The nuclear membrane features nuclear pores that allow the selective passage of molecules such as RNA and proteins. This selective permeability is crucial for processes like protein synthesis and gene expression, as it enables the necessary materials to enter and exit the nucleus efficiently.

In contrast, the cell membrane protects the entire cell and regulates the entry and exit of substances on a broader scale, while the cytoplasm encompasses all cellular components except the nucleus and is involved in various metabolic processes. The endoplasmic reticulum is a network involved in protein and lipid synthesis, but it does not specifically control the flow of materials to and from the nucleus. Thus, the nuclear membrane is the correct structure responsible for managing nuclear transport.