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Property: (right/left) serial
Definition: (right serial) $R$ is a direct sum of right ideals whose submodules are linearly ordered
Rings having property on left: 31
Rings not having property on left: 57
Rings having property on right: 29
Rings not having property on right: 59
Resources for learning about this property:
F. W. Anderson and K. R. Fuller. Rings and categories of modules. (2012) @ Chapter 8, Section 32
G. Puninski. Serial rings. (2001) @ .