Must the deceased husband's mother be informed of her share of the shop's rent each time, given that she refuses to take it and each time the questioner (the daughter-in-law) hears remarks, and is the questioner sinful if she does not inform her, knowing that she receives the amount periodically?

If the grandmother has appointed the questioner as her agent to collect her share of the shop's rent and disburse it to her grandchildren, there is no impediment to doing so after the collection, because the agent's possession is like the principal's possession. It is not necessary to inform the grandmother every time if the authorization is absolute. However, if the authorization is not absolute, then she must be informed every time the rent is collected.