In this paper, we show that, for the rational inner function , is reducible on the quotient module over the bidisk if and only if is the product of two one-variable inner functions.
"The reducibility of compressed shifts on a class of quotient modules over the bidisk." Ann. Funct. Anal. 10 (4) 447 - 459, November 2019. https://doi.org/10.1215/20088752-2019-0003