What is the ruling on a married man, whose wife does not live with him and he suspects she is having sexual relations, and he does not wish to divorce her, marrying another Muslim woman according to Sharia?

If the woman is a Muslim, it is not permissible to accuse her of anything that contradicts purity and chastity unless it is proven by decisive evidence, because the default for Muslims is integrity. If she is a woman of the Book (Jewish or Christian), she should not be accused either, because the default for marrying her is purity unless evidence proves otherwise. If she is neither Muslim nor a woman of the Book, marriage to her is invalid, according to the Almighty's saying: "And do not marry polytheistic women until they believe." Regarding marrying another woman in addition to her, there is no harm in it, as the Sharia permits a Muslim man to marry up to four women, provided there is justice and the ability to fulfill their rights.