matt grime said:Firstly you shoulnd't have r on both sides of that equality. Secondly, you will need to know the definition of 'primary'. So what is it? What does it imply? Does this seem familiar to any other result you know? Is the result even true? (i.e. think about what a counter example might be and see where it fails to satisfy some hypothesis.)
Irreducible components are the building blocks of a primary decomposition. They are the minimal nonempty closed subsets of a topological space that cannot be written as a union of two proper closed subsets.
Irreducible components correspond to primary ideals in a primary decomposition. Each primary ideal corresponds to a unique irreducible component, and vice versa.
Irreducible components play a crucial role in primary decomposition because they provide a way to decompose a complex topological space into simpler, more manageable parts. This allows for a better understanding and analysis of the space.
Yes, a topological space can have multiple primary decompositions. However, the number of irreducible components will remain the same in each decomposition.
The irreducible components in a primary decomposition can be determined using the Nullstellensatz theorem in algebraic geometry or the Prime Ideal Decomposition theorem in commutative algebra.