“After the wedding, I lived with my husband with his mother, in the block. When I meet older people I know from my immediate neighborhood, I bow to them. I consider it natural. However, my mother-in-law has a grudge against me. It is of the opinion, that I shouldn't bow to my neighbors, with whom she is at odds, because this is how I show a lack of loyalty to her. Which of us is right?”
I have no doubts, that it is you who follow the rules of good manners. I advise you to explain to your mother-in-law, that it is not in the interest of any of the inhabitants to spread disagreements and conflicts. On the other hand, a nice and well-mannered daughter-in-law is nice to show off to her surroundings. The evidence of cordiality and affection needed by the husband's mother can be shown in everyday situations.