In the name of God, the Most Merciful, the Most Just.
If you live in a city or in village then, I have a message for you that the Messenger of God said:
”Whoever acquires a dog - with the exception of a dog to guard livestock, a hunting dog, or a farm dog - each day a Qirat is deducted from his reward. [Abu Dawood (ref)]
I heard Allah's Apostle(P.B.U.H) saying; "Angels (of Mercy) do not enter a house wherein there is a dog or a picture of a living creature (a human being or an animal)." [Hadith - Bukhari 3:515 (ref)
It was narrated from Abu Hurayrah (may God be pleased with him) that the Messenger of God (peace and blessings of God be upon him) said: “The purification of the vessel of one of you, if a dog licks it, is to wash it seven times, the first time with soil.” Nararted by Muslim, 279.
And according to another report by Muslim (280):
”If a dog licks the vessel of one of you, let him wash it seven times and rub it with soil the eighth time.”
Shaykh al-Islam Ibn Taymiyah (may God have mercy on him) said:
With regard to dogs, there are three views among the scholars:
1 – That they are taahir (pure), even their saliva. This is the view of Maalik.
2 – That they are naajis (impure), even their hair. This is the view of al-Shaafa’i and is one of the two views narrated from Ahmad.
3 – Their hair is taahir but their saliva is naajis. This is the view of Abu Haneefah and of Ahmad in the other report narrated from him.
To be on the safe side it is better, if a person touches a dog and there is something wet on his hand, or if there is something wet on the dog, to wash his hand seven times, one of which should be with soil. Shaykh Ibn ‘Uthaymeen said:
”With regard to touching this dog, if there is no wetness then it does not make the hand naajis, but if he touches it and there is any wetness, then this means that the hand becomes naajis according to the view of many scholars, and the hand must be washed after that seven times, one of which should be with soil.
This is the most correct view. So if the wetness of the dog’s hair gets onto one’s garment or body, that does not make it naajis.”
My advice for you is to tell this to your mother that, dogs has its own purposes which Quran said.
The only thing that it is considered haraam in Islam is to grew up the dog inside the house because of its saliva. The saliva of dogs often contains microscopic worms, science has shown. Dogs are not to be kept inside the house at all times. They should be kept outside.
If you are unable to let the dog outside, then you must make a special place for him and when the prayer comes then you must keep the dog outside your room and only when the prayer is over then you can call him inside. If your mother lives in a city then
Shaykh Ibn ‘Uthaymeen (may God have mercy on him) said:
”Based on this, if a house is in the middle of the city there is no need to keep a dog to guard it, so keeping a dog for this purpose in such situations is haraam and is not permitted, and it detracts one or two qiraats from a person’s reward every day. They should get rid of this dog and not keep it. But if the house is in the countryside and there is no one else around, then it is permissible to keep a dog to guard the house and the people who are in it; guarding the members of the household is more important than guarding livestock or crops.”
And if your mother lives in a city or in a village then 2 good deeds will be cancelled and only one good deed is cancelled in countryside. And don’t worry, if your mother has bought a dog as a pet, then you won’t be held for responsibility on the Day of Judgment only your mother unless she gets rid of the dog and repent to God.