I figured it out. For months my telephone did not ring. I heard from people only via voice mail. I was, with some justice, suspected of being a recluse, a pensioned derelict. I searched the Samsung site and the T-Mobile site to no avail. It takes a password to reset everything and I didn't have one. I eventually found the factory preset password in the manual. Yet another example of computer security keeping the intended user out of a system no one is trying to break into. That was the manufacturer's portion of the stupidity.
For my part I was curious how to avoid this problem in the future. I finally found it in the T-Mobile site. One presses the mute button to toggle the ringer on and off and have it vibrate silently instead. Mute button? It has a mute button? Yes, one presses and holds the # key and the phone is muted. Worse, in addition to the # sign, the key actually has a tiny figure of an X-ed out speaker. That is the very icon that stubbornly refused to leave the tiny screen all these months. Like Poe's raven, it sat above my door and would leave, 'Nevermore'. What is so sad about this is that the icon on the button is recognizable if I hold it at just the right distance and am trying to focus on it. But this phone, and all phones, is designed for the youth market, people to whom that icon looks the size of a baked potato. Like they say in those movies about death row, "Ancient crock walking!"