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My understanding of this topic is that marriage is one man and one woman for life. According to 1 Cor 7:39-40 (see quote below) it is not a sin to marry if one's spouse has died. Personally I choose not to marry anyone who was Biblically married in the past.
Below are some bible verses on the topic in the order they are found in the Scriptures...
You shall not commit adultery.
Exo 20:14 (KJV)
And you shall not commit adultery.
Deu 5:18 (KJV)
If any man take a wife, and go in unto her, and hate her, And give occasions of speech against her, and bring up an evil name upon her, and say, I took this woman, and when I came to her, I found her not a maid: Then shall the father of the damsel, and her mother, take and bring forth the tokens of the damsel's virginity unto the elders of the city in the gate: And the damsel's father shall say unto the elders, I gave my daughter unto this man to wife, and he hateth her; And, lo, he hath given occasions of speech against her, saying, I found not thy daughter a maid; and yet these are the tokens of my daughter's virginity. And they shall spread the cloth before the elders of the city. And the elders of that city shall take that man and chastise him; And they shall amerce him in an hundred shekels of silver, and give them unto the father of the damsel, because he hath brought up an evil name upon a virgin of Israel: and she shall be his wife; he may not put her away all his days. But if this thing be true, and the tokens of virginity be not found for the damsel: Then they shall bring out the damsel to the door of her father's house, and the men of her city shall stone her with stones that she die: because she hath wrought folly in Israel, to play the whore in her father's house: so shalt thou put evil away from among you. If a man be found lying with a woman married to an husband, then they shall both of them die, both the man that lay with the woman, and the woman: so shalt thou put away evil from Israel. If a damsel that is a virgin be betrothed unto an husband, and a man find her in the city, and lie with her; Then ye shall bring them both out unto the gate of that city, and ye shall stone them with stones that they die; the damsel, because she cried not, being in the city; and the man, because he hath humbled his neighbour's wife: so thou shalt put away evil from among you. But if a man find a betrothed damsel in the field, and the man force her, and lie with her: then the man only that lay with her shall die: But unto the damsel thou shalt do nothing; there is in the damsel no sin worthy of death: for as when a man riseth against his neighbour, and slayeth him, even so is this matter: For he found her in the field, and the betrothed damsel cried, and there was none to save her. If a man find a damsel that is a virgin, which is not betrothed, and lay hold on her, and lie with her, and they be found; Then the man that lay with her shall give unto the damsel's father fifty shekels of silver, and she shall be his wife; because he hath humbled her, he may not put her away all his days.
Deu 22:13-29 (KJV)
"When a man takes a wife and marries her, if then she finds no favor in his eyes because he has found some indecency in her, and he writes her a certificate of divorce and puts it in her hand and sends her out of his house, and she departs out of his house, and if she goes and becomes another man's wife, and the latter man hates her and writes her a certificate of divorce and puts it in her hand and sends her out of his house, or if the latter man dies, who took her to be his wife, then her former husband, who sent her away, may not take her again to be his wife, after she has been defiled, for that is an abomination before the LORD. And you shall not bring sin upon the land that the LORD your God is giving you for an inheritance.
Deu 24:1-4 (KJV)
"It was also said, 'Whoever divorces his wife, let him give her a certificate of divorce.' But I say to you that everyone who divorces his wife, except on the ground of sexual immorality, makes her commit adultery, and whoever marries a divorced woman commits adultery.
Mat 5:31-32 (KJV)
And Pharisees came up and in order to test him asked, "Is it lawful for a man to divorce his wife?" He answered them, "What did Moses command you?" They said, "Moses allowed a man to write a certificate of divorce and to send her away." And Jesus said to them, "Because of your hardness of heart he wrote you this commandment. But from the beginning of creation, 'God made them male and female.' 'Therefore a man shall leave his father and mother and hold fast to his wife, and the two shall become one flesh.' So they are no longer two but one flesh. What therefore God has joined together, let not man separate." And in the house the disciples asked him again about this matter. And he said to them, "Whoever divorces his wife and marries another commits adultery against her, and if she divorces her husband and marries another, she commits adultery."
Mar 10:2-12 (KJV)
"Everyone who divorces his wife and marries another commits adultery, and he who marries a woman divorced from her husband commits adultery.
Luk 16:18 (KJV)
To the married I give this charge (not I, but the Lord): the wife should not separate from her husband (but if she does, she should remain unmarried or else be reconciled to her husband), and the husband should not divorce his wife.
1Co 7:10-11 (KJV
A wife is bound to her husband as long as he lives. But if her husband dies, she is free to be married to whom she wishes, only in the Lord. Yet in my judgment she is happier if she remains as she is. And I think that I too have the Spirit of God.
1Co 7:39-40 (KJV)
I only quote the below because (though this is off the topic of remarriage) some try to use the excuse that God divorced Israel to justify divorce (and even remarriage). God is faithful to Israel even though they are not always faithful to him.
Is God Divorced?The LORD said to me in the days of King Josiah: "Have you seen what she did, that faithless one, Israel, how she went up on every high hill and under every green tree, and there played the whore? And I thought, 'After she has done all this she will return to me,' but she did not return, and her treacherous sister Judah saw it. She saw that for all the adulteries of that faithless one, Israel, I had sent her away with a decree of divorce. Yet her treacherous sister Judah did not fear, but she too went and played the whore. Because she took her whoredom lightly, she polluted the land, committing adultery with stone and tree. Yet for all this her treacherous sister Judah did not return to me with her whole heart, but in pretense, declares the LORD." And the LORD said to me, "Faithless Israel has shown herself more righteous than treacherous Judah. Go, and proclaim these words toward the north, and say, "'Return, faithless Israel, declares the LORD. I will not look on you in anger, for I am merciful, declares the LORD; I will not be angry forever. Only acknowledge your guilt, that you rebelled against the LORD your God and scattered your favors among foreigners under every green tree, and that you have not obeyed my voice, declares the LORD. Return, O faithless children, declares the LORD; for I am your master; I will take you, one from a city and two from a family, and I will bring you to Zion.
Jer 3:6-1 (KJV)