Use a dutch oven or stock pot on medium high heat, add the vegetable oil and sausage and cook, stirring occasionally, until the sausage is browned, 6 minutes.
Lower the heat to medium and add the green pepper, onion, and celery. Cook until softened, 7 or 8 minutes. Add the garlic and cook an additional minute.
Add the canned tomatoes and spices. Rinse the tomato can with 1/2 cup of water and add to the pot. Bring to a boil.
To cook the rice in the same pot, add 3 cups of chicken broth, bring to a boil, then add the rice. Simmer for 15 minutes, adding additional broth if needed. Taste and adjust seasonings adding a small amount of cayenne pepper or Tabasco if desired.
Add the shrimp, stir and cook until pink, 4 minutes.
Serve the jambalaya hot over rice.