Is a wife considered divorced after a quarrel and the husband's anger and utterance of phrases such as "divorce is the solution" and "tell me one word: 'Divorce me,' so I can say to her, 'You are divorced,' " given that the wife did not respond or ask for divorce?

The phrases you uttered to your wife do not include an explicit pronouncement of divorce, such as "You are divorced." And your statement, "then divorce," is not explicit because it was not attributed to your wife. It is a condition for an explicit divorce that the pronouncement be attributed to the wife. Therefore, no divorce has occurred. It is advised to treat your wife with kindness and to be wary of anger, as it is the key to evil.