I have a question regarding the resulting table/relation from a union operation in relational algebra. Does the resulting relation have a primary key? Or does it return a table with a hidden primary key?

Example: Suppose we take the union of the following two tables:

    R1:
    id | name
    ----------
    1    a
    2    b
    3    c

    R2:
    id | name
    ----------
    1    b
    2    c
    3    d

Where the attribute "id" is primary key in both relation R1 and R2, now the union of those two will be:
    
    id | name
    ------------
    1     a
    1     b
    2     b
    2     c
    3     c
    3     d

Now none of the attributes in the resulting relation can be primary key since both id and name contain duplicates. How is this resolved by the DBMS?