Costochondritis: This condition, also known as Tietze's syndrome, involves inflammation of the cartilage that connects the ribs to the breastbone. It can cause sharp pains to the chest area, including the lateral ribs, and is often exacerbated by movement or deep breathing. The location and nature of the pain described make costochondritis a plausible explanation.

Intercostal Muscle Strain: Strain or injury to the muscles between the ribs can cause sharp, localized pain, especially if the strain occurs underneath the right arm. This could be due to overuse, direct injury, or sudden contraction of these muscles.
Rib Fracture: A fracture to one of the ribs could cause sharp pain, especially if the fracture is located in a position that corresponds to the area underneath the right arm. The pain would likely be exacerbated by movement, coughing, or deep breathing.
Precordial Catch Syndrome: This condition involves sudden, sharp pains in the chest or rib area, often precipitated by specific movements or positions. It's more common in adolescents and young adults but can occur at any age.

Pulmonary Embolism: Although less likely, a pulmonary embolism (a blood clot in the lungs) can cause sudden, sharp chest pain that might radiate to the side. This is a life-threatening condition that requires immediate medical attention.
Myocardial Infarction (Heart Attack): Chest pain is a classic symptom of a heart attack, and while the pain typically presents as a pressure or heaviness in the center of the chest, it can sometimes radiate to the sides or arms. It's crucial to consider this diagnosis, especially in individuals with risk factors.
Pneumothorax: A collapsed lung can cause sudden, sharp chest pain that worsens with deep breathing or coughing. The pain can be localized to one side and could be perceived as being underneath the arm if the collapse affects the upper part of the lung.

Slipping Rib Syndrome: This rare condition involves the slipping of the eighth, ninth, or tenth rib out of place, which can cause severe pain in the lower chest or upper abdominal area. The pain can sometimes be referred to the area underneath the arm.
Thoracic Outlet Syndrome: Compression of the nerves or blood vessels that pass through the thoracic outlet (the space between the collarbone and first rib) can cause pain in the chest, shoulder, or arm. While not typically the first consideration for sharp pain underneath the right arm, it's a possibility in some cases.
Sickle Cell Crisis: In individuals with sickle cell disease, a crisis can cause severe pain episodes, including chest pain, due to the sickling of red blood cells causing vascular occlusion. The pain can be sharp and localized to various areas, including the chest wall.