tinynerdi said:I forgot about that.. let try Z/ZxZ =~ Z Since ZxZ is a subring of Z.
A factor ring of a ring is a new ring that is created by taking a quotient of the original ring. This means that certain elements or subsets of the original ring are identified as equivalent and combined to form a new ring.
The factor ring of a ring is used to simplify and study the structure of a ring. It can also help to identify important properties of the original ring, such as the presence of certain ideals or subrings.
The factor ring of a ring is typically denoted as R/I, where R is the original ring and I is the ideal that is used to form the quotient. In some cases, a different notation, such as R//I, may be used.
In the context of a factor ring of a ring, an ideal is a subset of the original ring that is closed under addition, subtraction, and multiplication by elements of the ring. It is used to determine which elements or subsets of the original ring are identified as equivalent in the factor ring.
The factor ring of a ring is computed by first defining an ideal I in the original ring R. Then, the elements of I are used to create cosets, which are subsets of R that contain all elements of I plus one additional element. These cosets form the elements of the factor ring, with operations defined based on the original ring operations. This process is also known as taking the quotient of R by I.