What is the ruling on a man marrying a woman during her 'iddah (waiting period) after her husband's death, and what are the implications for this marriage that took place thirty years ago and in which they had children?

The woman whose husband has died is obligated to observe an 'iddah (waiting period) of four months and ten days, unless she is pregnant, in which case her 'iddah ends upon delivery. Marriage during the 'iddah is invalid by scholarly consensus. Therefore, a separation must occur between her and the man she married. She then completes the 'iddah from the first marriage, and once it is finished, the second man is permitted to marry her. Her children from the invalid marriage are attributed to him due to the شبهة (semblance of legality) arising from ignorance of the religious ruling.